Proof That Every Open Cover of a Closed and Bounded Interval is Reducible to a Finite Subcover

Theorem

Every open cover of a closed and bounded intervalis reducible to a finite subcover.

Proof

Ifis a closed and bounded interval with an open coverso that

Now apply the Heine - Borel theorem which states that a subspace of(with the usual topology) is compact if and only if it is closed and bounded. Sinceis closed and bounded it is compact hence has a finite subcover ofso that

for some