# Heine-Borel Theorem/Real Line

## Theorem

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

Let $C \subseteq \R$.

Then $C$ is closed and bounded in $\R$ if and only if $C$ is compact.

## Proof

### Necessary Condition

Let $C$ be closed and bounded in $\R$.

Then, by Closed Bounded Subset of Real Numbers is Compact, $C$ is compact.

$\Box$

### Sufficient Condition

Let $C$ be compact in $\R$.

Then, by Compact Subspace of Real Numbers is Closed and Bounded, $C$ is closed and bounded in $\R$.

$\blacksquare$

## Note

This does not apply in the general metric space.

A trivial example is $\left({0 \,.\,.\, 1}\right)$ as a subspace of itself.

It is closed and bounded but not compact.

## Source of Name

This entry was named for Heinrich Eduard Heine and Félix Édouard Justin Émile Borel.

The theorem is sometimes called the Borel-Lebesgue Theorem, for Émile Borel and Henri Léon Lebesgue.