Open Balls 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 open ball of $a$ of radius $\epsilon$.

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

Proof
Let $n \in \Z$.

Then:

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

From Leigh.Samphier/Sandbox/Closed Balls of P-adic Number:
 * $\map { B^{\,-}_{p^{-\paren{n + 1} } } } a = a + p^{n+1} \Z_p$

The result follows.