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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]] and let $A \\subseteq X$.\nIf $A$ with the [[subspace metric|0T2WYDTRP2]] is [[complete|08EWT7J0SC]], then $A$ is [[closed|0G5AV9Q4RR]] in $X$.\n\n## Remarks\n\nProof: let $x \\in \\overline{A}$. By [[closure is the set of limits of sequences|003QP7A8HG]], there is a sequence in $A$ with limit $x$. It is a Cauchy sequence in $A$, so it has a limit in $A$. Limits are unique in $X$, so $x \\in A$.\nThe space $X$ need not be complete here.\nFor example, $\\mathbb{Q}$ is not closed in $\\mathbb{R}$, so $\\mathbb{Q}$ is not complete.\nIn a complete space, a subset is complete if and only if it is closed, by [[closed subsets of complete spaces are complete|0FBXJZX3MK]].\n\n## Explanations\n\n### L1 human\n\nA complete subset contains the limits of its Cauchy sequences, and thus of its convergent sequences.\nA set that keeps all its limits is closed.\n",
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