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  "body": "\n## Statement\n\nLet $(a_n)$ be a real sequence.\nThen $(a_n)$ has a [[subsequence|0R8J3ZQMJS]] that is [[monotone|0KN22AAW5R]].\n\n## Remarks\n\nThe usual proof uses peaks. An index $n$ is a peak if $a_m \\le a_n$ for all $m \u003e n$.\nIf there are infinitely many peaks, the terms at the peaks form a decreasing subsequence.\nIf there are finitely many peaks, start after the last one. Each index then has a later index with a larger term, and this gives a strictly increasing subsequence.\nWith the [[monotone convergence theorem|0FM6N4R0VB]], the lemma gives the [[Bolzano–Weierstrass theorem|0KDJD249DT]].\n\n## Explanations\n\n### L1 human\n\nThink of the terms as heights of hills seen from left to right.\nEither infinitely many hills have no higher hill to their right, or from some point on you can always find a higher hill further right.\n",
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