Compact Subspace of Real Numbers is Closed and Bounded

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

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

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

Proof
From: the result follows by the Rule of Transposition.
 * Unbounded Set of Real Numbers is not Compact
 * Non-Closed Set of Real Numbers is not Compact