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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]] and let $K$ be a [[compact subset|00R0X1WGPX]] of $X$.\nThen $K$ is [[bounded|0P953MG05E]].\n\n## Remarks\n\nProof: if $K$ is empty, it is bounded. Otherwise fix $x \\in K$. The [[open balls|0DC3MPR7C9]] $B(x, n)$ for $n \\ge 1$ cover $K$, so finitely many of them do. The largest of these balls contains $K$.\nA compact subset of a metric space is also [[closed|0G5AV9Q4RR]], because metric spaces are Hausdorff.\nThe converse fails in general: a closed bounded set need not be compact. With the metric $\\min(|x - y|, 1)$, the real line is closed and bounded, but not compact.\n\n## Explanations\n\n### L1 human\n\nA compact set can be covered by finitely many balls about one point, and then it fits in the largest of them.\n",
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