Continuous Image of Compact Space is Compact/Corollary 3/Proof 1

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: Corollary 2, $f \left({S}\right)$ is bounded.

By Closure of Real Interval:
 * $\sup \left({f \left({S}\right)}\right) \in \operatorname {cl} \left({f \left({S}\right)}\right)$

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

By the above and Compact Subspace of Hausdorff Space is Closed, $f \left({S}\right)$ is closed in $\R$.

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

Hence the result.