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

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## 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:

the result follows by the Rule of Transposition.

$\blacksquare$

## Sources

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