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
Using the Rule of Transposition, it suffices to show that a compact subspace of $\R$ is closed.

This follows directly from:


 * Real Number Space Satisfies All Separation Axioms


 * Compact Subspace of Hausdorff Space is Closed