Heine–Borel Theorem

Heine–Borel Theorem: A subset $K\subseteq \mathbb{R}^k$ is compact (in the Euclidean metric) if and only if it is closed and bounded .

This theorem is the fundamental compactness criterion in Euclidean spaces and is used constantly to verify hypotheses of the extreme value theorem , uniform continuity , and convergence results.

Proof sketch (optional): If $K$ is compact, it is bounded (cover by balls of radius 1 and extract a finite subcover) and closed (limits of sequences in $K$ stay in $K$). Conversely, if $K$ is closed and bounded, any sequence in $K$ is bounded, hence has a convergent subsequence by Bolzano–Weierstrass ; closedness ensures the limit lies in $K$, giving sequential compactness and hence compactness.