Compact Subspace of Real Numbers is Closed and Bounded

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 1

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.

$\blacksquare$

Proof 2

From Real Number Line is Metric Space, $\left({\R, d}\right)$ is a metric space, where $d$ denotes the Euclidean metric on $\R$.

Therefore, the result follows from:

Metric Space is Hausdorff

Compact Subspace of Hausdorff Space is Closed

and:

Compact Metric Space is Totally Bounded

Totally Bounded Metric Space is Bounded

$\blacksquare$