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 $\R$ iff $C$ is compact.

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

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

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

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

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.

Also see

 * Bolzano-Weierstrass Theorem
 * Heine-Borel Theorem for Metric Spaces
 * Compact Subspace of Linearly Ordered Space

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