Sequential compactness equals compactness

Sequential compactness equals compactness: Let $(X,d)$ be a metric space and let $K\subseteq X$. Then $K$ is compact if and only if $K$ is sequentially compact , meaning every sequence in $K$ has a subsequence that is convergent to a point of $K$.

This equivalence fails in general topological spaces but is fundamental in metric topology and connects compactness to arguments using sequences.