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  "body": "\n## Statement\n\nLet $X$ be a [[compact|0QPHW703PV]] [[metric space|063W5JMY85]] and let $C(X)$ be the set of continuous functions $X \\to \\mathbb{R}$, with the [[uniform metric|06HE9J7DXF]] $d_\\infty(f, g) = \\sup_{x \\in X} |f(x) - g(x)|$.\nLet $\\mathcal{F} \\subseteq C(X)$.\nThen the closure of $\\mathcal{F}$ in $C(X)$ is compact if and only if $\\mathcal{F}$ is [[equicontinuous|0MW2WXTHMC]] and uniformly bounded.\nHere uniformly bounded means that there is a real $M$ with $|f(x)| \\le M$ for all $f \\in \\mathcal{F}$ and $x \\in X$.\n\n## Remarks\n\nA sequential form: each uniformly bounded and equicontinuous sequence of functions $[a, b] \\to \\mathbb{R}$ has a subsequence that converges uniformly on $[a, b]$.\nThe proof shows that $\\mathcal{F}$ is [[totally bounded|0WE5BBHGM6]] in the complete space $C(X)$, and then uses [[compact iff complete and totally bounded|054Q5PW5GX]].\nFor example, the functions $f \\colon [0, 1] \\to [0, 1]$ that are [[Lipschitz|0TMZ21WKCN]] with constant $1$ form a compact set in $C([0, 1])$.\nThe theorem is a main tool for existence proofs, for example in the Peano existence theorem for ordinary differential equations.\n\n## Explanations\n\n### L1 human\n\nIn $\\mathbb{R}^n$, bounded sequences have convergent subsequences. The Arzelà–Ascoli theorem gives the same result for sequences of functions, but you also need equicontinuity.\nEquicontinuity stops the functions from becoming more and more steep, as $x^n$ does near $1$.\n",
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