Continuous Function on Compact Subspace of Euclidean 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

An application of Continuous Function on Compact Space is Bounded.

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Compactness