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# a closed subset of a complete metric space is complete

Let $X$ be a complete metric space, and let $Y\subseteq X$ be a closed subset of $X$. Then $Y$ is complete.

Proof

Let $\{y_{n}\}\subseteq Y$ be a Cauchy sequence in $Y$. Then by the completeness of $X$, $y_{n}\rightarrow x$ for some $x\in X$. Then every neighborhood of $x$ contains points in $Y$, so $x\in\overline{Y}=Y$.

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