Closed Balls Centered on P-adic Number is Countable/Open Balls

Theorem
Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $a \in \Q_p$.

Then the set of all open balls centered on $a$ is the countable set:
 * $\mathcal B = \set{\map {B_{p^{-n}}} a : n \in \Z}$

Proof
Let $\epsilon \in \R_{\ge 0}$.