{"componentChunkName":"component---src-lekoarts-gatsby-theme-minimal-blog-core-templates-post-query-tsx","path":"/my-favorite-proofs-from-analysis-1","result":{"data":{"post":{"slug":"/my-favorite-proofs-from-analysis-1","title":"My favorite proofs from Analysis1","date":"December 19, 2019","tags":[{"name":"Math","slug":"math"}],"description":null,"body":"function _extends(){_extends=Object.assign||function(target){for(var i=1;i<arguments.length;i++){var source=arguments[i];for(var key in source){if(Object.prototype.hasOwnProperty.call(source,key)){target[key]=source[key];}}}return target;};return _extends.apply(this,arguments);}function _objectWithoutProperties(source,excluded){if(source==null)return{};var target=_objectWithoutPropertiesLoose(source,excluded);var key,i;if(Object.getOwnPropertySymbols){var sourceSymbolKeys=Object.getOwnPropertySymbols(source);for(i=0;i<sourceSymbolKeys.length;i++){key=sourceSymbolKeys[i];if(excluded.indexOf(key)>=0)continue;if(!Object.prototype.propertyIsEnumerable.call(source,key))continue;target[key]=source[key];}}return target;}function _objectWithoutPropertiesLoose(source,excluded){if(source==null)return{};var target={};var sourceKeys=Object.keys(source);var key,i;for(i=0;i<sourceKeys.length;i++){key=sourceKeys[i];if(excluded.indexOf(key)>=0)continue;target[key]=source[key];}return target;}/* @jsx mdx */var _frontmatter={\"title\":\"My favorite proofs from Analysis1\",\"date\":\"2019-12-19T16:33:00.000Z\",\"tags\":[\"Math\"],\"excerpt\":\"with intuitions\"};var makeShortcode=function makeShortcode(name){return function MDXDefaultShortcode(props){console.warn(\"Component \"+name+\" was not imported, exported, or provided by MDXProvider as global scope\");return mdx(\"div\",props);};};var layoutProps={_frontmatter:_frontmatter};var MDXLayout=\"wrapper\";return function MDXContent(_ref){var components=_ref.components,props=_objectWithoutProperties(_ref,[\"components\"]);return mdx(MDXLayout,_extends({},layoutProps,props,{components:components,mdxType:\"MDXLayout\"}),mdx(\"p\",null,\"I\\u2019ve encountered a lot of proofs from \",mdx(\"a\",_extends({parentName:\"p\"},{\"href\":\"https://www.mcgill.ca/study/2019-2020/courses/math-254\"}),\"MATH254\"),\", an introductory real analysis course. From my experience, most of the knowledge vanished the moment I walked out of the final exam room. But some proofs are just so elegant that forgetting them pains me.\"),mdx(\"p\",null,\"This post is a collection of the neatest ones from my perspective. Besides reconstructing proofs, I also try to explain them in my own words to help you and my future self follow the line of thought.\"),mdx(\"h2\",null,\"Triangle inequality (In \",mdx(\"span\",_extends({parentName:\"h2\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mi\",_extends({parentName:\"msup\"},{\"mathvariant\":\"double-struck\"}),\"R\"),mdx(\"mn\",{parentName:\"msup\"},\"2\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\mathbb{R}^2\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.8141079999999999em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"R\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.8141079999999999em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"2\"))))))))))),\")\"),mdx(\"p\",null,\"Probably the most important inequality in this course.\"),mdx(\"p\",null,mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"|x+y|\\\\leq|x|+|y|\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"))))),mdx(\"h3\",null,\"Proof\"),mdx(\"p\",null,\"First, we prove the claim: \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",{parentName:\"mrow\"},\"a\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\u2005\\u200A\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u27FA\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\u2005\\u200A\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"a\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",{parentName:\"mrow\"},\"a\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"|x|\\\\leq a \\\\iff -a\\\\leq x\\\\leq a\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.549em\",\"verticalAlign\":\"-0.024em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"a\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u27FA\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.7719400000000001em\",\"verticalAlign\":\"-0.13597em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"a\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.7719400000000001em\",\"verticalAlign\":\"-0.13597em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"a\"))))),mdx(\"span\",{\"className\":\"katex-display\"},mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mtext\",{parentName:\"mrow\"},\"Here\\xA0I\\xA0only\\xA0prove\\xA0\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2018\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2018\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u21D2\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\\"\"),mdx(\"mtext\",{parentName:\"mrow\"},\"direction.\"),mdx(\"mspace\",_extends({parentName:\"mrow\"},{\"linebreak\":\"newline\"})),mdx(\"mtext\",{parentName:\"mrow\"},\"The\\xA0other\\xA0direction\\xA0can\\xA0be\\xA0easily\\xA0obtained\\xA0by\\xA0\\\"opening\\\"\\xA0the\\xA0absolute\\xA0value.\"),mdx(\"mspace\",_extends({parentName:\"mrow\"},{\"linebreak\":\"newline\"})),mdx(\"mtable\",_extends({parentName:\"mrow\"},{\"rowspacing\":\"0.24999999999999992em\",\"columnalign\":\"right left\",\"columnspacing\":\"0em\"}),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"}))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mtext\",{parentName:\"mrow\"},\"Case1.\\xA0\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2265\"),mdx(\"mn\",{parentName:\"mrow\"},\"0\"))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"}))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",{parentName:\"mrow\"},\"a\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2265\"),mdx(\"mn\",{parentName:\"mrow\"},\"0\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u21D2\"),mdx(\"mi\",{parentName:\"mrow\"},\"a\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2265\"),mdx(\"mn\",{parentName:\"mrow\"},\"0\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u21D2\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"a\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mn\",{parentName:\"mrow\"},\"0\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mo\",_extends({parentName:\"mstyle\"},{\"lspace\":\"0em\",\"rspace\":\"0em\"}),\"\\u21D2\"))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"a\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",{parentName:\"mrow\"},\"a\"))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"}))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mtext\",{parentName:\"mrow\"},\"Case2.\\xA0\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mn\",{parentName:\"mrow\"},\"0\"))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"}))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",{parentName:\"mrow\"},\"a\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u21D2\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2265\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"a\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u21D2\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mn\",{parentName:\"mrow\"},\"0\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",{parentName:\"mrow\"},\"a\"))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mo\",_extends({parentName:\"mstyle\"},{\"lspace\":\"0em\",\"rspace\":\"0em\"}),\"\\u21D2\"))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"a\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",{parentName:\"mrow\"},\"a\")))))),mdx(\"mspace\",_extends({parentName:\"mrow\"},{\"linebreak\":\"newline\"}))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\text{Here I only prove } ``\\\\Rightarrow\\\" \\\\text{direction.}\\\\\\\\ \\\\text{The other direction can be easily obtained by \\\"opening\\\" the absolute value.}\\\\\\\\\\n\\\\begin{aligned}\\n&\\\\text{Case1. } x\\\\geq 0 \\\\\\\\\\n&x = |x|\\\\leq a, x\\\\geq0 \\\\Rightarrow a\\\\geq0\\\\Rightarrow -a \\\\leq 0\\\\leq x\\\\\\\\\\n\\\\Rightarrow &-a \\\\leq x \\\\leq a\\\\\\\\\\n&\\\\text{Case2. } x\\\\lt 0 \\\\\\\\\\n& -x =|x|\\\\leq a \\\\Rightarrow x\\\\geq-a\\\\Rightarrow x<0\\\\leq a\\\\\\\\\\n\\\\Rightarrow &-a \\\\leq x \\\\leq a\\\\\\\\\\n\\\\end{aligned}\\\\\\\\\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord text\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"Here\\xA0I\\xA0only\\xA0prove\\xA0\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2018\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2018\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u21D2\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\\"\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord text\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"direction.\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace newline\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord text\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"The\\xA0other\\xA0direction\\xA0can\\xA0be\\xA0easily\\xA0obtained\\xA0by\\xA0\\\"opening\\\"\\xA0the\\xA0absolute\\xA0value.\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace newline\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"9.000000000000002em\",\"verticalAlign\":\"-4.250000000000001em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mtable\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"col-align-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"4.750000000000001em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-6.910000000000001em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-5.41em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.9099999999999993em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u21D2\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.4099999999999993em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-0.9099999999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"0.5900000000000007em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u21D2\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"4.250000000000001em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"col-align-l\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"4.750000000000001em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-6.910000000000001em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord text\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"Case1.\\xA0\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2265\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"0\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-5.41em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"=\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"a\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2265\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u21D2\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"a\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2265\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u21D2\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"a\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.9099999999999993em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"a\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"a\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.4099999999999993em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord text\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"Case2.\\xA0\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"0\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-0.9099999999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"=\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"a\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u21D2\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2265\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"a\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u21D2\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"a\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"0.5900000000000007em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"a\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"a\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"4.250000000000001em\"}}),mdx(\"span\",{parentName:\"span\"})))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace newline\"}))))),mdx(\"p\",null,\"For the triangle inequality:\"),mdx(\"span\",{\"className\":\"katex-display\"},mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mtable\",_extends({parentName:\"semantics\"},{\"rowspacing\":\"0.24999999999999992em\",\"columnalign\":\"right left\",\"columnspacing\":\"0em\"}),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\")))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u21D2\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\")))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u21D2\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\")))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mo\",{parentName:\"mrow\"},\"\\u21D2\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\")))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mo\",{parentName:\"mrow\"},\"\\u21D2\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\")))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0by\\xA0the\\xA0claim\\xA0above\\xA0\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u25A1\")))))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\begin{aligned}\\n|x|&\\\\leq |x| \\\\Rightarrow -|x| \\\\leq x\\\\leq|x|\\\\\\\\\\n|y|&\\\\leq |y| \\\\Rightarrow -|y| \\\\leq y\\\\leq|y|\\\\\\\\\\n-|x|-|y|&\\\\leq x+y \\\\leq|x|+|y|\\\\\\\\\\n\\\\Rightarrow -(|x|+|y|)&\\\\leq x+y \\\\leq|x|+|y|\\\\\\\\\\n\\\\Rightarrow |x+y| &\\\\leq |x|+|y| \\\\text{ by the claim above} \\\\ \\\\square\\n\\\\end{aligned}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"7.500000000000002em\",\"verticalAlign\":\"-3.5000000000000018em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mtable\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"col-align-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"4em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-6.16em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-4.659999999999999em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.1599999999999984em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-1.6599999999999984em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u21D2\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-0.15999999999999837em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u21D2\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"3.5000000000000018em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"col-align-l\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"4em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-6.16em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u21D2\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-4.659999999999999em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u21D2\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.1599999999999984em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-1.6599999999999984em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-0.15999999999999837em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord text\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\xA0by\\xA0the\\xA0claim\\xA0above\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord amsrm\"}),\"\\u25A1\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"3.5000000000000018em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))))),mdx(\"h3\",null,\"Notes\"),mdx(\"p\",null,\"The very obvious \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"|x|\\\\leq|x|\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\")))),\" may be hard to come up with. When you have an equality, think about inequalities it may give you. The generalization for \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mi\",_extends({parentName:\"msup\"},{\"mathvariant\":\"double-struck\"}),\"R\"),mdx(\"mi\",_extends({parentName:\"msup\"},{\"mathvariant\":\"double-struck\"}),\"n\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\mathbb{R^n}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68889em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"R\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.664392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))))))))))),\" can be proved by induction.\"),mdx(\"h2\",null,mdx(\"span\",_extends({parentName:\"h2\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"double-struck\"}),\"Q\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\mathbb{Q}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.85556em\",\"verticalAlign\":\"-0.16667em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"Q\"))))),\" is dense in \",mdx(\"span\",_extends({parentName:\"h2\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"double-struck\"}),\"R\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\mathbb{R}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68889em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"R\")))))),mdx(\"p\",null,\"That is, for any \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2208\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"double-struck\"}),\"R\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2203\"),mdx(\"mi\",{parentName:\"mrow\"},\"q\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2208\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"double-struck\"}),\"Q\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x, y \\\\in \\\\mathbb{R}, x<y, \\\\exists q \\\\in \\\\mathbb{Q}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.7335400000000001em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2208\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.88333em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"R\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2203\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"q\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2208\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.85556em\",\"verticalAlign\":\"-0.16667em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"Q\"))))),\" such that \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mi\",{parentName:\"mrow\"},\"q\"),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x < q < y\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.5782em\",\"verticalAlign\":\"-0.0391em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.7335400000000001em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"q\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.625em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\")))),\" . Equivalently, there are infinite rational numbers between any two real numbers.\"),mdx(\"h3\",null,\"Proof\"),mdx(\"p\",null,\"Let \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"n\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2208\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"double-struck\"}),\"N\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mn\",{parentName:\"mrow\"},\"0\"),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mi\",{parentName:\"mfrac\"},\"n\")),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\u2005\\u200A\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u27FA\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\u2005\\u200A\"),mdx(\"mi\",{parentName:\"mrow\"},\"n\"),mdx(\"mo\",{parentName:\"mrow\"},\">\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mrow\",{parentName:\"mfrac\"},mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\")))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"n\\\\in\\\\mathbb{N},0<\\\\frac{1}{n} < y-x \\\\iff n>\\\\frac{1}{y-x}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.5782em\",\"verticalAlign\":\"-0.0391em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"n\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2208\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.88333em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"N\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.190108em\",\"verticalAlign\":\"-0.345em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.845108em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.6550000000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.394em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.345em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.7777700000000001em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.549em\",\"verticalAlign\":\"-0.024em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u27FA\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.5782em\",\"verticalAlign\":\"-0.0391em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"n\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\">\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.326216em\",\"verticalAlign\":\"-0.481108em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.845108em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.6550000000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin mtight\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.394em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.481108em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"})))))),\" by the \",mdx(\"a\",_extends({parentName:\"p\"},{\"href\":\"http://mathonline.wikidot.com/the-archimedean-property\"}),\"Archimedean property\"),\".\"),mdx(\"p\",null,\"Consider the set \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"S\"),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"{\"),mdx(\"mi\",{parentName:\"mrow\"},\"k\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2208\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"double-struck\"}),\"N\"),mdx(\"mo\",{parentName:\"mrow\"},\":\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"mfrac\"},\"k\"),mdx(\"mi\",{parentName:\"mfrac\"},\"n\")),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"}\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"S = \\\\{ k \\\\in \\\\mathbb{N}: \\\\frac{k}{n}\\\\lt  x \\\\}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.05764em\"}}),\"S\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"=\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"{\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03148em\"}}),\"k\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2208\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68889em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"N\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\":\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.2251079999999999em\",\"verticalAlign\":\"-0.345em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.8801079999999999em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.6550000000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.394em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.03148em\"}}),\"k\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.345em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\"}\")))),\". This set is finite, so its complement is infinite, non-empty especially. By the \",mdx(\"a\",_extends({parentName:\"p\"},{\"href\":\"https://brilliant.org/wiki/the-well-ordering-principle/\"}),\"well-ordering principle\"),\", the set \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"double-struck\"}),\"R\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2216\"),mdx(\"mi\",{parentName:\"mrow\"},\"S\"),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"{\"),mdx(\"mi\",{parentName:\"mrow\"},\"k\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2208\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"double-struck\"}),\"N\"),mdx(\"mo\",{parentName:\"mrow\"},\":\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"mfrac\"},\"k\"),mdx(\"mi\",{parentName:\"mfrac\"},\"n\")),mdx(\"mo\",{parentName:\"mrow\"},\">\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"}\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\mathbb{R}\\\\setminus S = \\\\{k\\\\in\\\\mathbb{N}:\\\\frac{k}{n}>x\\\\}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"R\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2216\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.05764em\"}}),\"S\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"=\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"{\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03148em\"}}),\"k\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2208\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68889em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"N\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\":\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.2251079999999999em\",\"verticalAlign\":\"-0.345em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.8801079999999999em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.6550000000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.394em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.03148em\"}}),\"k\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.345em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\">\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\"}\")))),\" has a least element.\"),mdx(\"p\",null,\"Let \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"m\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"m\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"m\")))),\" be the smallest natural number with \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"mfrac\"},\"m\"),mdx(\"mi\",{parentName:\"mfrac\"},\"n\")),mdx(\"mo\",{parentName:\"mrow\"},\">\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\frac{m}{n} > x\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.040392em\",\"verticalAlign\":\"-0.345em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.695392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.6550000000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.394em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"m\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.345em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\">\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\")))),\", especially \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mrow\",{parentName:\"mfrac\"},mdx(\"mi\",{parentName:\"mrow\"},\"m\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mn\",{parentName:\"mrow\"},\"1\")),mdx(\"mi\",{parentName:\"mfrac\"},\"n\")),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\frac{m-1}{n}< x\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.190108em\",\"verticalAlign\":\"-0.345em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.845108em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.6550000000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.394em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"m\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin mtight\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.345em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\")))),\". \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"m\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mi\",{parentName:\"mrow\"},\"n\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2208\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"double-struck\"}),\"N\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"mfrac\"},\"m\"),mdx(\"mi\",{parentName:\"mfrac\"},\"n\")),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2208\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"double-struck\"}),\"Q\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"m, n \\\\in \\\\mathbb{N}, \\\\frac{m}{n} \\\\in \\\\mathbb{Q}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.7335400000000001em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"m\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"n\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2208\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.040392em\",\"verticalAlign\":\"-0.345em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"N\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.695392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.6550000000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.394em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"m\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.345em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2208\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.85556em\",\"verticalAlign\":\"-0.16667em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"Q\"))))),\".\"),mdx(\"p\",null,mdx(\"strong\",{parentName:\"p\"},\"Claim:\"),\" \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"mfrac\"},\"m\"),mdx(\"mi\",{parentName:\"mfrac\"},\"n\")),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x<\\\\frac{m}{n}\\\\leq y\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.5782em\",\"verticalAlign\":\"-0.0391em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.040392em\",\"verticalAlign\":\"-0.345em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.695392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.6550000000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.394em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"m\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.345em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.625em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\")))),\" is the number we want.\"),mdx(\"span\",{\"className\":\"katex-display\"},mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mtable\",_extends({parentName:\"semantics\"},{\"rowspacing\":\"0.24999999999999992em\",\"columnalign\":\"right left\",\"columnspacing\":\"0em\"}),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"}))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"mfrac\"},\"m\"),mdx(\"mi\",{parentName:\"mfrac\"},\"n\")),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mrow\",{parentName:\"mfrac\"},mdx(\"mi\",{parentName:\"mrow\"},\"m\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mn\",{parentName:\"mrow\"},\"1\")),mdx(\"mi\",{parentName:\"mfrac\"},\"n\")),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mi\",{parentName:\"mfrac\"},\"n\")),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"}))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"mfrac\"},\"m\"),mdx(\"mi\",{parentName:\"mfrac\"},\"n\")),mdx(\"mo\",{parentName:\"mrow\"},\">\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0by\\xA0our\\xA0choice\\xA0of\\xA0m\"))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mo\",_extends({parentName:\"mstyle\"},{\"lspace\":\"0em\",\"rspace\":\"0em\"}),\"\\u21D2\"))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"mfrac\"},\"m\"),mdx(\"mi\",{parentName:\"mfrac\"},\"n\")),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u25A1\")))))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\begin{aligned}\\n&\\\\frac{m}{n} =\\\\frac{m-1}{n} + \\\\frac{1}{n} < x+y-x =y\\\\\\\\\\n&\\\\frac{m}{n} >x \\\\text{ by our choice of m}\\\\\\\\\\n\\\\Rightarrow &\\\\  x<\\\\frac{m}{n}<y \\\\ \\\\square\\n\\\\end{aligned}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"6.49456em\",\"verticalAlign\":\"-2.99728em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mtable\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"col-align-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"3.49728em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-5.497279999999999em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.4037200000000003em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-1.3101600000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u21D2\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"2.99728em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"col-align-l\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"3.49728em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-5.497279999999999em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"1.10756em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.314em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"n\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.677em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"m\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.686em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"=\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"1.32144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.314em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"n\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.677em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"m\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"1\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.686em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"1.32144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.314em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"n\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.677em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"1\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.686em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"=\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.4037200000000003em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"1.10756em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.314em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"n\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.677em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"m\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.686em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\">\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord text\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\xA0by\\xA0our\\xA0choice\\xA0of\\xA0m\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-1.3101600000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"1.10756em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.314em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"n\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.677em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"m\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.686em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord amsrm\"}),\"\\u25A1\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"2.99728em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))))),mdx(\"h3\",null,\"Notes\"),mdx(\"p\",null,\"The choice of \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mi\",{parentName:\"mfrac\"},\"n\")),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\frac{1}{n}<y-x\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.190108em\",\"verticalAlign\":\"-0.345em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.845108em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.6550000000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.394em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.345em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.7777700000000001em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\")))),\" in the first step seems mysterious, but everything follows that is easy to understand. The reason why we made this choice is: \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mi\",{parentName:\"mfrac\"},\"n\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\frac{1}{n}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.190108em\",\"verticalAlign\":\"-0.345em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.845108em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.6550000000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.394em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.345em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"})))))),\" is the step size, if it\\u2019s greater than \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"y\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"y-x\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.7777700000000001em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\")))),\", assume \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"z\"),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mi\",{parentName:\"mrow\"},\"z\"),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mi\",{parentName:\"mfrac\"},\"n\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"z<x, z+\\\\frac{1}{n}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.5782em\",\"verticalAlign\":\"-0.0391em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.04398em\"}}),\"z\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.7777700000000001em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.04398em\"}}),\"z\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.190108em\",\"verticalAlign\":\"-0.345em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.845108em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.6550000000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.394em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.345em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"})))))),\" may be greater than \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"y\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"y\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.625em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\")))),\". That is, we may go past \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"y\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"y\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.625em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\")))),\" directly, which is bad for finding a number \",mdx(\"strong\",{parentName:\"p\"},\"between\"),\" \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mi\",{parentName:\"mrow\"},\"y\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x, y\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.625em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03588em\"}}),\"y\")))),\".\"),mdx(\"p\",null,\"Of course, the set \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"S\"),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"{\"),mdx(\"mi\",{parentName:\"mrow\"},\"k\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2208\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"double-struck\"}),\"N\"),mdx(\"mo\",{parentName:\"mrow\"},\":\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"mfrac\"},\"k\"),mdx(\"mi\",{parentName:\"mfrac\"},\"n\")),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"}\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"S =\\\\{k\\\\in\\\\mathbb{N}:\\\\frac{k}{n} \\\\lt x\\\\}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.05764em\"}}),\"S\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"=\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"{\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03148em\"}}),\"k\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2208\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68889em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"N\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\":\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.2251079999999999em\",\"verticalAlign\":\"-0.345em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.8801079999999999em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.6550000000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.394em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.03148em\"}}),\"k\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.345em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\"}\")))),\" may be empty ( \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mi\",{parentName:\"mfrac\"},\"n\")),mdx(\"mo\",{parentName:\"mrow\"},\">\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\frac{1}{n}\\\\gt x\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.190108em\",\"verticalAlign\":\"-0.345em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.845108em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.6550000000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.394em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.345em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\">\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\")))),\" ), but that doesn\\u2019t matter. \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"double-struck\"}),\"R\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2216\"),mdx(\"mi\",{parentName:\"mrow\"},\"S\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\mathbb{R}\\\\setminus S\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"R\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2216\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.05764em\"}}),\"S\")))),\" is \\u201Cmore\\u201D infinite. The well-ordering principle holds as long as \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"double-struck\"}),\"R\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2216\"),mdx(\"mi\",{parentName:\"mrow\"},\"S\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\mathbb{R}\\\\setminus S\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"R\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2216\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.05764em\"}}),\"S\")))),\" is non-empty. Using infinity to show non-emptiness may seem like an overkill, but interestingly, this kind of \\u201Coverkill\\u201D happens all the time. It\\u2019s easier to get information from a stronger statement.\"),mdx(\"blockquote\",null,mdx(\"p\",{parentName:\"blockquote\"},\"We require f to be continuous on the whole domain, but actually we only used continuity at a single point in this proof. \\u2014 My analysis professor\")),mdx(\"h2\",null,\"Every continuous function on a compact domain is uniformly continuous.\"),mdx(\"h3\",null,\"Proof\"),mdx(\"p\",null,\"Let \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",{parentName:\"mrow\"},\":\"),mdx(\"mi\",{parentName:\"mrow\"},\"A\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2192\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"double-struck\"}),\"R\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"f: A\\u2192 \\\\mathbb{R}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\":\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"A\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2192\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68889em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"R\"))))),\" be continuous. Let \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"\\u03F5\"),mdx(\"mo\",{parentName:\"mrow\"},\">\"),mdx(\"mn\",{parentName:\"mrow\"},\"0\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\epsilon  >0,\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.5782em\",\"verticalAlign\":\"-0.0391em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"\\u03F5\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\">\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.8388800000000001em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\")))),\" then \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2200\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2208\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mi\",{parentName:\"mrow\"},\"A\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2203\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03B4\"),mdx(\"mi\",{parentName:\"msub\"},\"x\")),mdx(\"mo\",{parentName:\"mrow\"},\">\"),mdx(\"mn\",{parentName:\"mrow\"},\"0\"),mdx(\"mo\",{parentName:\"mrow\"},\":\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"u\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03B4\"),mdx(\"mi\",{parentName:\"msub\"},\"x\")),mdx(\"mo\",{parentName:\"mrow\"},\"\\u27F9\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mi\",{parentName:\"mrow\"},\"u\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mn\",{parentName:\"mfrac\"},\"2\")),mdx(\"mi\",{parentName:\"mrow\"},\"\\u03F5\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\forall \\\\ x\\\\in \\\\ A,\\\\ \\\\exists \\\\delta _{x}  >0:|x-u|< \\\\delta _{x} \\\\Longrightarrow \\\\ |f( \\\\ x) \\\\ -f( \\\\ u) |\\\\ < \\\\ \\\\frac{1}{2} \\\\epsilon\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.73354em\",\"verticalAlign\":\"-0.0391em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2200\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2208\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"A\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2203\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03785em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\">\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.64444em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\":\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.84444em\",\"verticalAlign\":\"-0.15em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03785em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u27F9\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.190108em\",\"verticalAlign\":\"-0.345em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.845108em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.6550000000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"2\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.394em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.345em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"\\u03F5\")))),\" (1)\"),mdx(\"p\",null,\"Consider the neighborhoods \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"V\"),mdx(\"mrow\",{parentName:\"msub\"},mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mn\",{parentName:\"mfrac\"},\"2\")),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03B4\"),mdx(\"mi\",{parentName:\"msub\"},\"x\")))),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2200\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2208\"),mdx(\"mi\",{parentName:\"mrow\"},\"A\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"V_{\\\\frac{1}{2} \\\\delta _{x} }( x) ,\\\\ \\\\forall x\\\\in A\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.23702em\",\"verticalAlign\":\"-0.48701999999999995em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.22222em\"}}),\"V\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.34480000000000005em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.7537800000000003em\",\"marginLeft\":\"-0.22222em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter sizing reset-size3 size6\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.8443142857142858em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.656em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"2\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.2255000000000003em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line mtight\",\"style\":{\"borderBottomWidth\":\"0.049em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.384em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.344em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter sizing reset-size3 size6\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.16454285714285719em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginLeft\":\"-0.03785em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.48701999999999995em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2200\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2208\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"A\")))),\". Then, \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"C\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mo\",{parentName:\"mrow\"},\":\"),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"{\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"V\"),mdx(\"mrow\",{parentName:\"msub\"},mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mn\",{parentName:\"mfrac\"},\"2\")),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03B4\"),mdx(\"mi\",{parentName:\"msub\"},\"x\")))),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mo\",{parentName:\"mrow\"},\":\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2208\"),mdx(\"mi\",{parentName:\"mrow\"},\"A\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"}\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"C\\\\ :=\\\\{V_{\\\\frac{1}{2} \\\\delta _{x} }( x) :x\\\\in A\\\\}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.07153em\"}}),\"C\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\":\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.36687em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"=\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.23702em\",\"verticalAlign\":\"-0.48701999999999995em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"{\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.22222em\"}}),\"V\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.34480000000000005em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.7537800000000003em\",\"marginLeft\":\"-0.22222em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter sizing reset-size3 size6\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.8443142857142858em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.656em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"2\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.2255000000000003em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line mtight\",\"style\":{\"borderBottomWidth\":\"0.049em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.384em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.344em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter sizing reset-size3 size6\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.16454285714285719em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginLeft\":\"-0.03785em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.48701999999999995em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\":\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.5782em\",\"verticalAlign\":\"-0.0391em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2208\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"A\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\"}\")))),\" is an open cover of A since an arbitrary union of open sets are still open. A is compact, so \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"C\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"C\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.07153em\"}}),\"C\")))),\" has a finite subcover, \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"U\"),mdx(\"mo\",{parentName:\"mrow\"},\":\"),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"{\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"V\"),mdx(\"mrow\",{parentName:\"msub\"},mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mn\",{parentName:\"mfrac\"},\"2\")),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03B4\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"k\"))))),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mn\",{parentName:\"msub\"},\"1\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u22EF\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\u2009\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"V\"),mdx(\"mrow\",{parentName:\"msub\"},mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mn\",{parentName:\"mfrac\"},\"2\")),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03B4\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"k\"))))),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"k\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"}\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"U:=\\\\{V_{\\\\frac{1}{2} \\\\delta _{x_{k} }}( x_{1}) ,\\\\ \\\\cdots ,V_{\\\\frac{1}{2} \\\\delta _{x_{k} }}( x_{k})\\\\}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10903em\"}}),\"U\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\":\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.36687em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"=\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.27114em\",\"verticalAlign\":\"-0.5211399999999999em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"{\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.22222em\"}}),\"V\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3448em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.75378em\",\"marginLeft\":\"-0.22222em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter sizing reset-size3 size6\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.8443142857142858em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.656em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"2\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.2255000000000003em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line mtight\",\"style\":{\"borderBottomWidth\":\"0.049em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.384em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.344em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter sizing reset-size3 size6\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.1645428571428572em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.3569999999999998em\",\"marginLeft\":\"-0.03785em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3448em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.3448em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.1em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.69444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.03148em\"}}),\"k\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.34963999999999995em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.39274285714285706em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.5211399999999999em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"minner\"}),\"\\u22EF\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.22222em\"}}),\"V\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3448em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.75378em\",\"marginLeft\":\"-0.22222em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter sizing reset-size3 size6\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.8443142857142858em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.656em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"2\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.2255000000000003em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line mtight\",\"style\":{\"borderBottomWidth\":\"0.049em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.384em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.344em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter sizing reset-size3 size6\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.1645428571428572em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.3569999999999998em\",\"marginLeft\":\"-0.03785em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3448em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.3448em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.1em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.69444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.03148em\"}}),\"k\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.34963999999999995em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.39274285714285706em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.5211399999999999em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.33610799999999996em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.03148em\"}}),\"k\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\"}\")))),\" where \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mn\",{parentName:\"msub\"},\"1\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u22EF\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\u2009\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"k\")),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2208\"),mdx(\"mi\",{parentName:\"mrow\"},\"A\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x_{1} ,\\\\cdots ,x_{k} \\\\in A\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.7335400000000001em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"minner\"}),\"\\u22EF\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.33610799999999996em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.03148em\"}}),\"k\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2208\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"A\"))))),mdx(\"p\",null,\"Let \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mi\",{parentName:\"mrow\"},\"u\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2208\"),mdx(\"mi\",{parentName:\"mrow\"},\"A\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x,\\\\ u\\\\ \\\\in A\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.7335400000000001em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2208\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"A\")))),\" be arbitrary. We want to show \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"u\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mi\",{parentName:\"mrow\"},\"\\u03B4\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u27F9\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"u\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mi\",{parentName:\"mrow\"},\"\\u03F5\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"|x-u|< \\\\delta \\\\Longrightarrow |f( x) -f( u) |< \\\\epsilon\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.71844em\",\"verticalAlign\":\"-0.024em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u27F9\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"\\u03F5\")))),\". For now, we only know \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"f\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"f\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\")))),\" behaves nicely locally(within each neighborhood around \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mn\",{parentName:\"mrow\"},\"1\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2A7D\"),mdx(\"mi\",{parentName:\"mrow\"},\"i\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2A7D\"),mdx(\"mi\",{parentName:\"mrow\"},\"k\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x_{i} ,\\\\ 1\\\\leqslant i\\\\leqslant k\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.8388800000000001em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"1\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel amsrm\"}),\"\\u2A7D\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.79619em\",\"verticalAlign\":\"-0.13667em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"i\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel amsrm\"}),\"\\u2A7D\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03148em\"}}),\"k\")))),\" ), but uniformly continuity is a global property.\"),mdx(\"p\",null,mdx(\"em\",{parentName:\"p\"},\"Intuition\"),\": If \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mi\",{parentName:\"mrow\"},\"u\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x, u\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.625em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\")))),\" are in the same neighborhood around \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x_{i}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.58056em\",\"verticalAlign\":\"-0.15em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\", we can utilize the nice, local behavior. The problem is that \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03B4\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\")))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\delta _{x_{i} }\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.9445399999999999em\",\"verticalAlign\":\"-0.2501em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03785em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3280857142857143em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.2501em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\" depends on \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x_{i}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.58056em\",\"verticalAlign\":\"-0.15em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\", so the required distance varies in different locations on the domain. Can we find a \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"\\u03B4\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\delta\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\")))),\" that makes \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"x\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\")))),\" and \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"u\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"u\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\")))),\" \\u201Cclose enough\\u201D to be considered locally, no matter where they are?\"),mdx(\"p\",null,\"Yes, we can. Why? Because \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"U\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"U\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10903em\"}}),\"U\")))),\" is finite.\"),mdx(\"p\",null,\"Let \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"\\u03B4\"),mdx(\"mo\",{parentName:\"mrow\"},\":\"),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mi\",{parentName:\"mrow\"},\"min\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2061\"),mdx(\"mrow\",{parentName:\"mrow\"},mdx(\"mo\",_extends({parentName:\"mrow\"},{\"fence\":\"true\"}),\"{\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mn\",{parentName:\"mfrac\"},\"2\")),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03B4\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mn\",{parentName:\"msub\"},\"1\"))),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u22EF\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\u2009\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mn\",{parentName:\"mfrac\"},\"2\")),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03B4\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"k\"))),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"fence\":\"true\"}),\"}\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\delta :=\\\\min\\\\left\\\\{\\\\frac{1}{2} \\\\delta_{{x}_{1}},\\\\cdots ,\\\\frac{1}{2} \\\\delta_{{x}_{k}}\\\\right\\\\}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\":\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.36687em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"=\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.20001em\",\"verticalAlign\":\"-0.35001em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mop\"}),\"min\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"minner\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen delimcenter\",\"style\":{\"top\":\"0em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"delimsizing size1\"}),\"{\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.845108em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.6550000000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"2\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.394em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.345em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03785em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31731428571428577em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.2501em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"minner\"}),\"\\u22EF\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.845108em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.6550000000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"2\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.394em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.345em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.55em\",\"marginLeft\":\"-0.03785em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3448em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.3487714285714287em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.03148em\"}}),\"k\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15122857142857138em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.25586em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose delimcenter\",\"style\":{\"top\":\"0em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"delimsizing size1\"}),\"}\")))))),\". \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"U\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"U\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10903em\"}}),\"U\")))),\" covers A, so \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2208\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"V\"),mdx(\"mrow\",{parentName:\"msub\"},mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mn\",{parentName:\"mfrac\"},\"2\")),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03B4\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\"))))),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x \\\\in V_{\\\\frac{1}{2} \\\\delta_{{x}_{i}}}( x_{i})\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.5782em\",\"verticalAlign\":\"-0.0391em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2208\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.2536800000000001em\",\"verticalAlign\":\"-0.50368em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.22222em\"}}),\"V\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3448em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.75378em\",\"marginLeft\":\"-0.22222em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter sizing reset-size3 size6\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.8443142857142858em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.656em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"2\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.2255000000000003em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line mtight\",\"style\":{\"borderBottomWidth\":\"0.049em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.384em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.344em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter sizing reset-size3 size6\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.1645428571428572em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginLeft\":\"-0.03785em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3448em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.3448em\",\"marginRight\":\"0.1em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.65952em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31472em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3678em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.50368em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\")))),\"for some \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\")),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2208\"),mdx(\"mi\",{parentName:\"mrow\"},\"A\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x_{i} \\\\in A\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.6891em\",\"verticalAlign\":\"-0.15em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2208\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"A\")))),\".\"),mdx(\"span\",{\"className\":\"katex-display\"},mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mtable\",_extends({parentName:\"semantics\"},{\"rowspacing\":\"0.24999999999999992em\",\"columnalign\":\"center\",\"columnspacing\":\"1em\"}),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mtable\",_extends({parentName:\"mstyle\"},{\"rowspacing\":\"0.24999999999999992em\",\"columnalign\":\"right left\",\"columnspacing\":\"0em\"}),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\")),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\")))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mn\",{parentName:\"mfrac\"},\"2\")),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03B4\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\"))),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03B4\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\"))),mdx(\"mo\",{parentName:\"mrow\"},\"\\u27F9\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mn\",{parentName:\"mfrac\"},\"2\")),mdx(\"mi\",{parentName:\"mrow\"},\"\\u03F5\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0by\\xA0(1)\"))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"u\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\")),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\")))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"u\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\")),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mtext\",{parentName:\"mstyle\"},\"triangle\\xA0ineq\"))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"u\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\")),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mi\",{parentName:\"mrow\"},\"\\u03B4\"),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mn\",{parentName:\"mfrac\"},\"2\")),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03B4\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\"))),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mn\",{parentName:\"mfrac\"},\"2\")),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03B4\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\"))),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mn\",{parentName:\"mfrac\"},\"2\")),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03B4\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\"))),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03B4\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\"))))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"u\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\")),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03B4\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\")))))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mo\",{parentName:\"mrow\"},\"\\u27F9\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"u\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mn\",{parentName:\"mfrac\"},\"2\")),mdx(\"mi\",{parentName:\"mrow\"},\"\\u03F5\"))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"u\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\")))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"u\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"}))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2264\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"u\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"}))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",_extends({parentName:\"mtd\"},{\"scriptlevel\":\"0\",\"displaystyle\":\"true\"}),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mrow\",{parentName:\"mrow\"}),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mn\",{parentName:\"mfrac\"},\"2\")),mdx(\"mi\",{parentName:\"mrow\"},\"\\u03F5\"),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mn\",{parentName:\"mfrac\"},\"2\")),mdx(\"mi\",{parentName:\"mrow\"},\"\\u03F5\"),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mi\",{parentName:\"mrow\"},\"\\u03F5\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u25A1\")))))))))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\begin{gathered}\\n\\\\begin{aligned}\\n|x-x_{i} | & < \\\\frac{1}{2} \\\\delta _{ {x}_{i} } < \\\\delta _{ {x}_{i} } \\\\Longrightarrow |f( x) -f( x_{i}) |< \\\\frac{1}{2} \\\\epsilon \\\\ \\\\text{by (1)}\\\\\\\\\\n|u-x_{i} | & =|u-x+x-x_{i} |\\\\\\\\\\n \\\\text{triangle ineq} & \\\\leq |u-x|+|x-x_{i} |< \\\\delta +\\\\frac{1}{2} \\\\delta _{ {x}_{i} } \\\\leq \\\\frac{1}{2} \\\\delta _{ {x}_{i} } +\\\\frac{1}{2} \\\\delta _{ {x}_{i} } =\\\\delta _{ {x}_{i} }\\\\\\\\\\n|u-x_{i} |< \\\\delta _{ {x}_{i} }& \\\\Longrightarrow |f( u) -f( x_{i}) |< \\\\frac{1}{2} \\\\epsilon \\\\\\\\\\n|f( x) -f( u) | & =|f( x) -f( x_{i}) +f( x_{i}) -f( u) |\\\\\\\\\\n & \\\\leq |f( x) -f( x_{i}) |+|f( x_{i}) -f( u) |\\\\\\\\\\n & < \\\\frac{1}{2} \\\\epsilon +\\\\frac{1}{2} \\\\epsilon =\\\\epsilon \\\\ \\\\square \\\\\\\\\\n\\n\\\\end{aligned}\\\\\\\\\\n\\\\end{gathered}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"14.029760000000003em\",\"verticalAlign\":\"-6.764880000000001em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mtable\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"col-align-c\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"7.2648800000000024em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-9.264880000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"9.114880000000001em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mtable\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"col-align-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"7.114880000000001em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-9.114880000000003em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-7.288880000000001em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-5.30744em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord text\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"triangle\\xA0ineq\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.000000000000001em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03785em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3280857142857143em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.2501em\"}}),mdx(\"span\",{parentName:\"span\"})))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-1.1740000000000004em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"0.3259999999999996em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"2.307440000000001em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"6.614880000000001em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"col-align-l\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"7.114880000000001em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-9.114880000000003em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"1.32144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.314em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"2\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.677em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"1\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.686em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03785em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3280857142857143em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.2501em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03785em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3280857142857143em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.2501em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u27F9\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"1.32144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.314em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"2\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.677em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"1\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.686em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"\\u03F5\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord text\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"by\\xA0(1)\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-7.288880000000001em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"=\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-5.30744em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"1.32144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.314em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"2\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.677em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"1\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.686em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03785em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3280857142857143em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.2501em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"1.32144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.314em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"2\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.677em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"1\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.686em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03785em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3280857142857143em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.2501em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"1.32144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.314em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"2\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.677em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"1\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.686em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03785em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3280857142857143em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.2501em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"=\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03785em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3280857142857143em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.2501em\"}}),mdx(\"span\",{parentName:\"span\"})))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.000000000000001em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u27F9\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"1.32144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.314em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"2\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.677em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"1\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.686em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"\\u03F5\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-1.1740000000000004em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"=\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"0.3259999999999996em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2264\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"2.307440000000001em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3.32144em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"1.32144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.314em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"2\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.677em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"1\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.686em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"\\u03F5\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"+\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"1.32144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.314em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"2\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.677em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"1\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.686em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"\\u03F5\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"=\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"\\u03F5\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord amsrm\"}),\"\\u25A1\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"6.614880000000001em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"6.764880000000001em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))))),mdx(\"h3\",null,\"Notes\"),mdx(\"p\",null,\"If \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"U\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"U\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10903em\"}}),\"U\")))),\" is infinite, we are unable to take the minimum of all neighborhoods. To recap, our goal is to find a \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"\\u03B4\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\delta\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\")))),\" that is \\u201Cclose enough\\u201D for arbitrary \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mi\",{parentName:\"mrow\"},\"u\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2208\"),mdx(\"mi\",{parentName:\"mrow\"},\"A\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x,\\\\ u\\\\ \\\\in A\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.7335400000000001em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2208\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"A\")))),\" such that \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"u\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2223\"),mdx(\"mo\",{parentName:\"mrow\"},\"<\"),mdx(\"mi\",{parentName:\"mrow\"},\"\\u03F5\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"|f( x) -f( u) |< \\\\epsilon\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mbin\"}),\"\\u2212\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"<\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"\\u03F5\")))),\". The idea is to use \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x_{i}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.58056em\",\"verticalAlign\":\"-0.15em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\" as a bridge between \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"x\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\")))),\" and \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"u\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"u\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\")))),\" since we only know how \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"f\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"f\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\")))),\" behaves near each \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x_{i}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.58056em\",\"verticalAlign\":\"-0.15em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\". Choosing \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},\"1\"),mdx(\"mn\",{parentName:\"mfrac\"},\"2\")),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03B4\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\")))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\frac{1}{2} \\\\delta _{ {x}_{i} }\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.190108em\",\"verticalAlign\":\"-0.345em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen nulldelimiter\"})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mfrac\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.845108em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.6550000000000002em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"2\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.23em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.394em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.345em\"}}),mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose nulldelimiter\"}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03785em\"}}),\"\\u03B4\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03785em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3280857142857143em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.2501em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\"neighborhoods ensures \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"x\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\xA0\"),mdx(\"mi\",{parentName:\"mrow\"},\"u\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x,\\\\ u\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.625em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\")))),\" are in the same neighborhood of \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"i\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x_{i}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.58056em\",\"verticalAlign\":\"-0.15em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"i\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\". This enables us to rely on \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"f\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"f\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10764em\"}}),\"f\")))),\"\\u2019s local behavior.\"),mdx(\"h2\",null,\"Bolzano-Weierstrass Theorem: Every bounded sequence in \",mdx(\"span\",_extends({parentName:\"h2\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"double-struck\"}),\"R\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\mathbb{R}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68889em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"R\"))))),\" has a convergent subsequence\"),mdx(\"p\",null,\"The proof is only for \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"double-struck\"}),\"R\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\mathbb{R}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68889em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"R\"))))),\", but the theorem also holds in \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mi\",_extends({parentName:\"msup\"},{\"mathvariant\":\"double-struck\"}),\"R\"),mdx(\"mi\",_extends({parentName:\"msup\"},{\"mathvariant\":\"double-struck\"}),\"n\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\mathbb{R^n}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68889em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"R\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.664392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))))))))))),\" and \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mi\",_extends({parentName:\"msup\"},{\"mathvariant\":\"double-struck\"}),\"C\"),mdx(\"mi\",{parentName:\"msup\"},\"n\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\mathbb{C}^n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68889em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathbb\"}),\"C\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.664392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\"))))))))))),\".\"),mdx(\"p\",null,mdx(\"strong\",{parentName:\"p\"},\"Definition:\"),\" Let \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"n\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"(x_n)\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\")))),\" be a sequence of real numbers. \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"N\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x_N\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.58056em\",\"verticalAlign\":\"-0.15em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.32833099999999993em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.10903em\"}}),\"N\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\" is called a \",mdx(\"em\",{parentName:\"p\"},\"peak\"),\" if \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2200\"),mdx(\"mi\",{parentName:\"mrow\"},\"n\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2265\"),mdx(\"mi\",{parentName:\"mrow\"},\"N\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"N\")),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2265\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"n\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\forall n\\\\geq N, x_N\\\\geq x_n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.83041em\",\"verticalAlign\":\"-0.13597em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2200\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"n\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2265\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.8777699999999999em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10903em\"}}),\"N\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.32833099999999993em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.10903em\"}}),\"N\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2265\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.58056em\",\"verticalAlign\":\"-0.15em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\" i.e., \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"N\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x_N\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.58056em\",\"verticalAlign\":\"-0.15em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.32833099999999993em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.10903em\"}}),\"N\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\" is the largest term in the sequence starting from it.\"),mdx(\"h3\",null,\"Lemma: Every sequence of real numbers has a monotone subsequence\"),mdx(\"h4\",null,\"Proof\"),mdx(\"p\",null,\"Let \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"n\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"(x_n)\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\")))),\" be a sequence of real numbers. There are two cases regrading the number of peaks:\"),mdx(\"p\",null,mdx(\"strong\",{parentName:\"p\"},\"Case1:\"),\" \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"n\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"(x_n)\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\")))),\" has infinitely many peaks. We can find a subsequence \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"n\"),mdx(\"mi\",{parentName:\"msub\"},\"k\"))),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"(x_{n_{k} })\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.00586em\",\"verticalAlign\":\"-0.25586em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.55em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3448em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.3487714285714287em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.03148em\"}}),\"k\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15122857142857138em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.25586em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\")))),\" consisting of peaks. \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"n\"),mdx(\"mn\",{parentName:\"msub\"},\"1\"))),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"(x_{n_{1} })\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.0001em\",\"verticalAlign\":\"-0.2501em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31731428571428577em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.2501em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\")))),\" is a peak, so \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"n\"),mdx(\"mn\",{parentName:\"msub\"},\"1\"))),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2265\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"n\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2200\"),mdx(\"mi\",{parentName:\"mrow\"},\"n\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2265\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"n\"),mdx(\"mn\",{parentName:\"msub\"},\"1\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x_{n_{1} } \\\\geq x_n, \\\\forall n\\\\geq n_1\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.88607em\",\"verticalAlign\":\"-0.2501em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31731428571428577em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.2501em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2265\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2200\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"n\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2265\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.58056em\",\"verticalAlign\":\"-0.15em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"n\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\". Thus, \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"n\"),mdx(\"mn\",{parentName:\"msub\"},\"1\"))),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2265\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"n\"),mdx(\"mn\",{parentName:\"msub\"},\"2\")))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x_{n_{1} }\\\\geq x_{n_{2} }\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.88607em\",\"verticalAlign\":\"-0.2501em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31731428571428577em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"1\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.2501em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2265\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68066em\",\"verticalAlign\":\"-0.2501em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31731428571428577em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"2\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.2501em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\". Similarly, \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"n\"),mdx(\"mn\",{parentName:\"msub\"},\"2\")))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x_{n_{2} }\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68066em\",\"verticalAlign\":\"-0.2501em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31731428571428577em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"2\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.2501em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\" is a peak, so \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"n\"),mdx(\"mn\",{parentName:\"msub\"},\"2\"))),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2265\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"n\"),mdx(\"mn\",{parentName:\"msub\"},\"3\"))),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2265\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"n\"),mdx(\"mn\",{parentName:\"msub\"},\"4\"))),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2265\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u22EF\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x_{n_{2} } \\\\geq x_{n_{3} } \\\\geq x_{n_{4} } \\\\geq \\\\cdots\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.88607em\",\"verticalAlign\":\"-0.2501em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31731428571428577em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"2\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.2501em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2265\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.88607em\",\"verticalAlign\":\"-0.2501em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31731428571428577em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"3\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.2501em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2265\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.88607em\",\"verticalAlign\":\"-0.2501em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.31731428571428577em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.357em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"4\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.143em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.2501em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2265\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.31em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"minner\"}),\"\\u22EF\")))),\", etc.\"),mdx(\"p\",null,\"The sequence is monotone increasing.\"),mdx(\"p\",null,mdx(\"strong\",{parentName:\"p\"},\"Case2: \"),\" \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"n\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"(x_n)\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\")))),\" has finitely many peaks. Then, we choose our subsequence after the last peak. Let \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"N\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x_N\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.58056em\",\"verticalAlign\":\"-0.15em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.32833099999999993em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.10903em\"}}),\"N\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\" be the last peak. For \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"n\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mi\",{parentName:\"mrow\"},\"n\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2265\"),mdx(\"mi\",{parentName:\"mrow\"},\"N\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2203\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"m\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mi\",{parentName:\"mrow\"},\"m\"),mdx(\"mo\",{parentName:\"mrow\"},\">\"),mdx(\"mi\",{parentName:\"mrow\"},\"n\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mtext\",{parentName:\"mrow\"},\"such\\xA0that\\xA0\\xA0\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"m\")),mdx(\"mo\",{parentName:\"mrow\"},\">\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"n\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x_n, n\\\\geq N, \\\\exists x_m, m\\\\gt n, \\\\text{such that } \\\\ x_m \\\\gt x_n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.8304100000000001em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"n\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\"\\u2265\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.10903em\"}}),\"N\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2203\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"m\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"m\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\">\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"n\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord text\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"such\\xA0that\\xA0\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"m\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\">\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.58056em\",\"verticalAlign\":\"-0.15em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\" since \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"n\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x_n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.58056em\",\"verticalAlign\":\"-0.15em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\" is not a peak. Similarly, \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"m\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"x_m\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.58056em\",\"verticalAlign\":\"-0.15em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"m\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\" is not a peak, so \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u2203\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"k\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mi\",{parentName:\"mrow\"},\"k\"),mdx(\"mo\",{parentName:\"mrow\"},\">\"),mdx(\"mi\",{parentName:\"mrow\"},\"m\"),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"separator\":\"true\"}),\",\"),mdx(\"mtext\",{parentName:\"mrow\"},\"such\\xA0that\\xA0\\xA0\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"k\")),mdx(\"mo\",{parentName:\"mrow\"},\">\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"m\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\exists x_k, k\\\\gt m, \\\\text{such that } \\\\ x_k \\\\gt x_m\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2203\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.33610799999999996em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.03148em\"}}),\"k\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\",\"style\":{\"marginRight\":\"0.03148em\"}}),\"k\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\">\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"m\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord text\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"such\\xA0that\\xA0\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.33610799999999996em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.03148em\"}}),\"k\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mrel\"}),\">\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.58056em\",\"verticalAlign\":\"-0.15em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"m\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\". Following this procedure, we end up with a monotone increasing subsequence.\"),mdx(\"p\",null,\"In both cases, there exists a monotone subsequence in \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"n\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"(x_n)\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\")))),\", so the lemma is proved.\"),mdx(\"h3\",null,\"Proof of the theorem\"),mdx(\"p\",null,mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"n\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"(x_n)\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\")))),\" is bounded. By lemma, \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"mi\",{parentName:\"msub\"},\"n\")),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"(x_n)\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\")))),\" has a monotone subsequence \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\"(\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"x\"),mdx(\"msub\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"msub\"},\"n\"),mdx(\"mi\",{parentName:\"msub\"},\"k\"))),mdx(\"mo\",_extends({parentName:\"mrow\"},{\"stretchy\":\"false\"}),\")\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"(x_{n_{k} })\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"1.00586em\",\"verticalAlign\":\"-0.25586em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mopen\"}),\"(\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"x\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15139199999999997em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.55em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"n\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.3448em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.3487714285714287em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.07142857142857144em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.5em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size3 size1 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\",\"style\":{\"marginRight\":\"0.03148em\"}}),\"k\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15122857142857138em\"}}),mdx(\"span\",{parentName:\"span\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.25586em\"}}),mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mclose\"}),\")\")))),\". By the \",mdx(\"a\",_extends({parentName:\"p\"},{\"href\":\"http://mathonline.wikidot.com/the-monotone-convergence-theorem/\"}),\"monotone convergence theorem\"),\", the sequence converges \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",_extends({parentName:\"mrow\"},{\"mathvariant\":\"normal\"}),\"\\u25A1\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\square\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.675em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord amsrm\"}),\"\\u25A1\")))),\".\"),mdx(\"h3\",null,\"Notes\"),mdx(\"p\",null,\"The proof is elegant and more importantly, easy to follow. The definition of \",mdx(\"em\",{parentName:\"p\"},\"peak\"),\" is the key observation which makes the proof intuitive.\"),mdx(\"blockquote\",null,mdx(\"p\",{parentName:\"blockquote\"},\"The power of a perfect definition. \\u2014 The Teaching assistant\")));};MDXContent.isMDXComponent=true;","excerpt":"with intuitions","banner":null}},"pageContext":{"slug":"/my-favorite-proofs-from-analysis-1"}}}