Non-Closed Set of Real Numbers is not Compact/Proof 2

Theorem

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

Let $S \subseteq \R$ be non-closed in $\R$.

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

Proof

From:

Real Number Line is Metric Space

Metric Space is Hausdorff

Compact Subspace of Hausdorff Space is Closed

the result follows by the rule of transposition.

$\blacksquare$