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  "body": "\n## Proof\n\n1. Let $(X, d)$ be a [[complete metric space|08EWT7J0SC]] and let $F$ be a [[closed subset|0G5AV9Q4RR]] of $X$. Let $d_F$ be the [[subspace metric|0T2WYDTRP2]] on $F$.\n2. Let $(x_n)$ be a [[Cauchy sequence|0MCK8K2D7Z]] in $(F, d_F)$.\n3. We have $d(x_m, x_n) = d_F(x_m, x_n)$ for all $m, n$. So $(x_n)$ is also a Cauchy sequence in $(X, d)$.\n4. Since $X$ is complete, $(x_n)$ [[converges|06JYPJ4WHH]] in $X$ to some point $x \\in X$.\n5. All $x_n$ lie in $F$, and $F$ is closed. Since [[closed sets contain their limits|09GQ3ZTQJ9]], we get $x \\in F$.\n6. For each $n$, $d_F(x_n, x) = d(x_n, x)$. So $d_F(x_n, x) \\to 0$, and $(x_n)$ converges to $x$ in $(F, d_F)$.\n7. Thus each Cauchy sequence in $F$ converges to a point of $F$, and $(F, d_F)$ is complete.\n\n## Idea\n\nA Cauchy sequence in $F$ has a limit in the big space, because the big space is complete.\nA closed set cannot lose a limit of its own points, so the limit lies in $F$.\n",
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