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

Proof

Let $x \in \overline Y$ be a point in the closure of $Y$ Then by the definition of closure, from each ball $B(x, \frac 1 n)$ centered in $x$ we can select a point $y_n \in Y$ This is clearly a Cauchy sequence in $Y$ and its limit is $x$ hence by the completeness of $Y$ $x \in Y$ and thus $Y = \overline Y$