Compact iff complete and totally bounded

Let $(X,d)$ be a metric space and let $K\subseteq X$ with the subspace metric.

Theorem: The following are equivalent:

• $K$ is compact .
• $K$ is complete and totally bounded .

This theorem is the fundamental metric characterization of compactness: “no missing limits” (completeness) plus “finite $\varepsilon$-approximation at every scale” (total boundedness).