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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]].\nThen $X$ is [[complete|08EWT7J0SC]] if and only if each [[Cauchy sequence|0MCK8K2D7Z]] in $X$ has a [[convergent|06JYPJ4WHH]] subsequence.\n\n## Remarks\n\nThe key fact: if a Cauchy sequence $(x_n)$ has a subsequence that converges to $x$, then $x_n \\to x$. For $\\varepsilon \u003e 0$, choose $N$ for $\\varepsilon / 2$ in the Cauchy condition and a term $x_{n_k}$ with $n_k \\ge N$ and $d(x_{n_k}, x) \u003c \\varepsilon / 2$.\nA consequence: a metric space in which each sequence has a convergent subsequence is complete. Thus each [[sequentially compact|0CARW59WD2]] metric space is complete.\n\n## Explanations\n\n### L1 human\n\nTo show that a space is complete, you need not find the limit of a whole Cauchy sequence.\nIt is enough to find a limit of some subsequence, and then the whole sequence follows it.\n",
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