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.

$\blacksquare$


Sources

  • 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Compactness
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