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

Jump to navigation
Jump to search

## 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:

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

$\blacksquare$