Unbounded Set of Real Numbers is not Compact/Proof 1

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

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

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

Proof
Let $\mathcal C$ be the set of all open $\epsilon$-balls centered at $0 \in \R$:
 * $\mathcal C = \left\{{B_\epsilon \left({0}\right): \epsilon \in \R_{>0}}\right\}$

We have that:
 * $\displaystyle \bigcup \mathcal C = \R \subseteq S$

Thus $\mathcal C$ is an open cover of $S$.

Let $\mathcal F$ be a finite subcover of $\mathcal C$ for $S$.

Then $\displaystyle \bigcup \mathcal F$ is the largest open $\epsilon$-ball in $\mathcal F$

But since $S$ is not bounded in $\R$:
 * $S \nsubseteq \displaystyle \bigcup \mathcal F$

Thus an open cover of $S$ has been demonstrated which has no finite subcover.

So, by definition, $S$ is not a compact subspace of $\R$.