Unbounded Set of Real Numbers is not Compact/Proof 1

Proof
By the rule of transposition, it suffices to show that if $S$ is a compact subspace of $\R$, then $S$ is bounded.

Let $\CC$ be the set of all open $\epsilon$-balls of $0$ in $\R$:
 * $\CC = \set {\map {B_\epsilon} 0: \epsilon \in \R_{>0}}$

We have that:
 * $\ds \bigcup \CC = \R \supseteq S$

From Open Ball of Metric Space is Open Set, it follows that $\CC$ is an open cover for $S$.

Let $\FF$ be a finite subcover of $\CC$ for $S$.

Then $\ds \bigcup \FF$ is the largest open $\epsilon$-ball in $\FF$.

Thus:
 * $S \subseteq \ds \bigcup \FF \in \CC$

Hence, $S$ is bounded.