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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]] and let $F$ be a [[closed subset|0G5AV9Q4RR]] of $X$.\nLet $(x_n)$ be a sequence of points of $F$ that [[converges|06JYPJ4WHH]] to a point $x \\in X$.\nThen $x \\in F$.\n\n## Remarks\n\nProof: if $x \\notin F$, then $X \\setminus F$ is open and contains $x$. Thus some ball $B(x, \\varepsilon)$ misses $F$. But $x_n \\in B(x, \\varepsilon)$ for large $n$, a contradiction.\nIn a metric space the converse is also true: a set that contains the limits of all its convergent sequences is closed. This follows from [[closure is the set of limits of sequences|003QP7A8HG]].\nThe statement holds in every [[topological space|0CNQXVBHCX]], with the same proof.\n\n## Explanations\n\n### L1 human\n\nYou cannot leave a closed set by a limit of a sequence.\nFor example, $1/n \\to 0$ in $\\mathbb{R}$, and $0$ is not in $(0, 1]$, which shows that $(0, 1]$ is not closed.\n",
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