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  "body": "\n## Statement\n\nLet $(a_n)$ be a [[bounded sequence|05FQ3VGAR9]] of real numbers.\nThen $(a_n)$ has a [[subsequence|0R8J3ZQMJS]] that converges to a real number.\n\n## Remarks\n\nThe proof takes a [[monotone subsequence|0MHGJRZMTM]]. It is bounded, so it converges by the [[monotone convergence theorem|0FM6N4R0VB]].\nThe limit of the subsequence lies in each closed interval $[m, M]$ that contains all terms, by [[the preservation of non-strict inequalities|0N32V0E3XJ]].\nThe theorem holds in $\\mathbb{R}^k$ for each $k$. It says that each closed and bounded subset of $\\mathbb{R}^k$ is sequentially compact.\nThe theorem fails in $\\mathbb{Q}$. The decimal approximations of $\\sqrt{2}$ are bounded, and no subsequence has a limit in $\\mathbb{Q}$.\n\n## History\n\nBernard Bolzano proved the result in 1817, as a lemma for the intermediate value theorem. Karl Weierstrass proved it again about fifty years later and made it a central tool of analysis.\n\n## Explanations\n\n### L1 human\n\nA bounded sequence cannot run away, so its terms must crowd together somewhere.\nThe theorem picks infinitely many terms that approach one point of crowding.\nFor $(-1)^n$, the even terms form a subsequence that converges to $1$.\n",
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