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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]], let $S \\subseteq X$ and let $x \\in X$.\nThen $x$ is in the [[closure|0DRKT1QJR7]] $\\overline{S}$ if and only if there is a sequence $(s_n)$ of points of $S$ that [[converges|06JYPJ4WHH]] to $x$.\n\n## Remarks\n\nProof of the direct part: by [[closure in a metric space|068P0P32DW]], for each $n \\ge 1$ there is $s_n \\in S$ with $d(x, s_n) \u003c 1/n$. Then $s_n \\to x$.\nThe converse follows because $\\overline{S}$ is closed and [[closed sets contain their limits|09GQ3ZTQJ9]].\nA consequence: $S$ is closed if and only if the limit of each convergent sequence of points of $S$ is in $S$.\nIn a general topological space, sequences are not enough, and one uses nets or filters instead.\n\n## Explanations\n\n### L1 human\n\nIn a metric space, the closure of a set consists of the limits of sequences that lie in the set.\nFor example, each real number is a limit of rational numbers, so the closure of $\\mathbb{Q}$ is $\\mathbb{R}$.\n",
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