Closed Ball in Euclidean Space is Compact

Theorem
Let $x \in \R_n$ be a point in the Euclidean space $\R^n$.

Let $\epsilon \in \R_{>0}$.

Then the closed $\epsilon$-ball $B_\epsilon^- \left({x}\right)$ is compact.

Proof
From Closed Ball is Closed, it follows that $B_\epsilon^- \left({x}\right)$ is closed in $\R^n$.

For all $a \in B_\epsilon^- \left({x}\right)$ we have $d \left({x, a}\right) \le \epsilon$, where $d$ denotes the Euclidean metric.

Then $B_\epsilon^- \left({x}\right)$ is bounded in $\R^n$.

From the Heine-Borel Theorem, it follows that $B_\epsilon^- \left({x}\right)$ is compact.