Compactness criteria in R^k

Let $K\subseteq \mathbb{R}^k$ with the Euclidean metric .

Proposition (equivalent characterizations): The following are equivalent:

• $K$ is compact (every open cover has a finite subcover).
• $K$ is sequentially compact (every sequence in $K$ has a convergent subsequence with limit in $K$).
• $K$ is closed and bounded .

This packages together the key compactness theorems specialized to Euclidean spaces (Bolzano–Weierstrass + Heine–Borel + compactness–sequential compactness equivalence).

Proof sketch: Compact $\Leftrightarrow$ sequentially compact holds in all metric spaces. Closed and bounded $\Rightarrow$ sequentially compact follows from Bolzano–Weierstrass plus closedness. Sequentially compact $\Rightarrow$ bounded and closed follows by applying sequential compactness to sequences escaping to infinity (for boundedness) and to convergent sequences (for closedness).