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

Corollary to Continuous Image of Compact Space is Compact

Let $S$ be a compact topological space.

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

Then $f$ attains its bounds on $S$.

Proof

By Continuous Image of Compact Space is Compact, $f \sqbrk S$ is compact.

From Compact Metric Space is Complete and Compact Metric Space is Totally Bounded, $f \sqbrk S$ is complete and totally bounded.

A Totally Bounded Metric Space is Bounded.

Hence both the supremum and the infimum of $f \sqbrk S$ exist in $\R$.

Because $f \sqbrk S$ is complete:

$\sup f \sqbrk S \in f \sqbrk S$

and:

$\inf f \sqbrk S \in f \sqbrk S$

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$\blacksquare$