Continuous Real-Valued Function on Compact Space is Bounded

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Theorem

Let $\struct {K, \tau}$ be a compact space.

Let $\R$ denote the real number line.

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

Then:

$f$ is bounded.

Proof

From Compact Space is Pseudocompact:

$\struct {K, \tau}$ is pseudocompact

By definition of pseudocompact:

$f$ is bounded.

$\blacksquare$

Sources

1960: Leonard Gillman and Meyer Jerison: Rings of Continuous Functions: Chapter $1$: Functions on a Topological Space, $\S 1.4$