Continuous Function on Compact Space is Bounded

Theorem
Let $\R^n$ be the $n$-dimensional Euclidean space.

Let $S \subseteq \R^n$ be a compact subspace of $\R^n$.

Let $f: S \to \R$ be a continuous function.

Then $f$ is bounded in $\R$.

Proof (based on the Heini-Borel Theorem)
TODO: