Closed Real Interval is Compact/Metric Space

Theorem
Let $\R$ be the real number line considered as an Euclidean space.

Let $I = \closedint a b$ be a closed real interval.

Then $I$ is compact.

Proof
From Closed Real Interval is Closed Set, $I$ is a closed set of $\R$.

From Real Interval is Bounded in Real Numbers, $I$ is bounded in $\R$.

The result follows by definition of compact.

Also see

 * Open Real Interval is not Compact