(Theorem) A closed subspace of a complete metric space is complete

Theorem: A closed subspace of a complete metric space is complete

If $(X,d)$ is a complete metric space and $A\subseteq X$ is closed, then $(A,d{|}_{A\times A})$ is complete.

Proof from Topology.

Let $(x_n)$ be a Cauchy sequence in $A$. Then $x_n$ converges to some $x\in X$, and since $A$ is closed, we have $x\in \overline{A}=A$.