Closed subsets of complete spaces are complete

Statement

Let $(X, d)$ be a complete metric space and let $F$ be a closed subset of $X$. Then $F$ with the subspace metric is complete.

Remarks

Proof: let $(x_n)$ be a Cauchy sequence in $F$. It is Cauchy in $X$, so it converges to some $x \in X$. Then $x \in F$, because closed sets contain their limits. For example, $[0, 1]$, $\mathbb{Z}$ and each closed ball of $\mathbb{R}^n$ are complete. The converse is complete subspaces are closed.

Explanations

L1 human

A Cauchy sequence in $F$ has a limit in the big complete space. Because $F$ is closed, this limit cannot escape from $F$.