Open Ball in P-adic Numbers is Closed Ball

Theorem
Let $p$ be a prime number.

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

Let $a \in \Q_p$.

For all $\epsilon \in \R_{>0}$:
 * Let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$
 * Let $\map {B^-_\epsilon} a$ denote the closed $\epsilon$-ball of $a$.

Then:
 * $\forall n \in Z : \map {B_{p^{-n} } } a = \map {B^-_{p^{-\paren {n + 1} } } } a$

Proof
Let $n \in \Z$.

Then:

By set equality:
 * $\map {B_{p^{-n} } } a = \map {B^-_{p^{-\paren {n + 1} } } } a$