Unbounded Set of Real Numbers is not Compact/Proof 2

Theorem

Let $\R$ be the set of real numbers considered as a Euclidean space.

Let $S \subseteq \R$ be unbounded in $\R$.

Then $S$ is not a compact subspace of $\R$.

Proof

From:

• Real Number Line is Metric Space
• Compact Metric Space is Totally Bounded
• Totally Bounded Metric Space is Bounded

the result follows by the rule of transposition.

$\blacksquare$