## Proof That Every Bounded Closed Interval is Countably Compact

Theorem

Every bounded closed intervalis countably compact.

Proof

A subsetof a topological spaceis countably compact if every finite subsethas an accumulation point in

The Bolzano - Weierstrass Theorem states that every bounded infinite set of real numbers contains an accumulation point.

Thushas an accumulation pointSinceis cl;osed andthe accumulation pointofbelongs toHenceis countably compact.