Sequential compactness equals compactness (metric spaces)

Sequential compactness equals compactness: Let $(X,d)$ be a metric space and $K\subseteq X$. Then $K$ is compact (every open cover has a finite subcover) if and only if $K$ is sequentially compact (every sequence in $K$ has a convergent subsequence with limit in $K$).

This equivalence is special to metric (first countable) spaces and makes compactness usable via sequences, which is often the most practical viewpoint in analysis.