Compact Subspace of Real Numbers is Closed and Bounded/Proof 1

Theorem

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

Let $S \subseteq \R$ be compact subspace of $\R$.

Then $S$ is closed and bounded in $\R$.

Proof

From:

Non-Closed Set of Real Numbers is not Compact

Unbounded Set of Real Numbers is not Compact

the result follows by the Rule of Transposition.

$\blacksquare$

Sources

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