Closed Ball of P-adic Number

Theorem
Let $p$ be a prime number.

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

Let $\Z_p$ be the $p$-adic integers.

Let $a \in \Q_p$.

For all $\epsilon \in \R_{>0}$, let $\map {{B_\epsilon}^-} a$ denote the closed ball of $a$ of radius $\epsilon$.

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

Proof
Let $n \in \Z$.

Then:

From set equality, $\map {B^{\,-}_{p^{-n}}} a = a + p^n \Z_p$.

The result follows.