Countably Compact Space is Pseudocompact

Theorem
Let $T = \struct {S, \tau}$ be a countably compact space.

Then $T$ is a pseudocompact space.

Proof
Let $T = \struct {S, \tau}$ be a countably compact space.

Then every countable open cover of $S$ has a finite subcover.

By definition:
 * $T$ is pseudocompact every continuous real-valued function on $S$ is bounded.

Let $f: S \to \R$ be such that $f$ is continuous.

Consider the sets:
 * $S_n = \set {x \in S: \size {\map f x} < n}$

$S_n$ forms a countable cover of $S$ whose finite subcover yields a bound for the absolute value of $f$.

Hence the result.