Closed Real Interval is Compact

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.

Topological Space
This proves that $I$ is compact in the context of a general topological space:

Metric Space
This proves that $I$ is compact in the context of a metric space:

Normed Vector Space
This proves that $I$ is compact in the context of a normed vector space:

Also see

 * Open Real Interval is not Compact