Continuous Image of Compact Space is Compact/Corollary 3/Proof 2

Corollary to Continuous Image of Compact Space is Compact
Let $f: S \to \R$ be a real-valued function.

If $S$ is a compact space, then $f$ attains its bounds on $S$.

Proof
By Continuous Image of Compact Space is Compact, $f \left({S}\right)$ is compact.

By the Heine-Borel Theorem (General Case), $f \left({S}\right)$ is complete and totally bounded.

A Totally Bounded Metric Space is Bounded.

Hence the supremum $\sup f \left({S}\right)$ and the infimum $\inf f \left({S}\right)$ exist.

Because $f \left({S}\right)$ is complete:
 * $\sup f \left({S}\right) \in f \left({S}\right)$

and:
 * $\inf f \left({S}\right) \in f \left({S}\right)$