# 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.

$\blacksquare$