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 $X$, $y_n\to x$ for some $x\in X$. Then every neighborhood of $x$ contains points in $Y$, so $x\in \overline{Y}=Y$.