# Compact Subspace of Real Numbers is Closed and Bounded

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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 1

From:

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:

and:

$\blacksquare$