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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[complete metric space|08EWT7J0SC]] and let $F$ be a [[closed subset|0G5AV9Q4RR]] of $X$.\nThen $F$ with the [[subspace metric|0T2WYDTRP2]] is complete.\n\n## Remarks\n\nProof: let $(x_n)$ be a Cauchy sequence in $F$. It is Cauchy in $X$, so it converges to some $x \\in X$. Then $x \\in F$, because [[closed sets contain their limits|09GQ3ZTQJ9]].\nFor example, $[0, 1]$, $\\mathbb{Z}$ and each closed ball of $\\mathbb{R}^n$ are complete.\nThe converse is [[complete subspaces are closed|0W41XG4W60]].\n\n## Explanations\n\n### L1 human\n\nA Cauchy sequence in $F$ has a limit in the big complete space.\nBecause $F$ is closed, this limit cannot escape from $F$.\n",
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