Heine-Borel Theorem/Euclidean Space

Theorem
For any natural number $n \ge 1$, a subspace $C$ of the Euclidean space $\R^n$ is closed and bounded iff it is compact.

Sufficient Condition
Let $C \subseteq \R^n$ be compact.

From Compact Subspace of Metric Space is Bounded, it follows that $C$ is bounded.

From Metric Space is Hausdorff, it follows that $\R^n$ is a Hausdorff space.

Then Compact Subspace of Hausdorff Space is Closed shows that $C$ is closed.