Sphere is Disjoint Union of Open Balls in P-adic Numbers

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 $\map {S_\epsilon} a$ denote the sphere of $a$ of radius $\epsilon$.


 * let $\map {B_\epsilon} a$ denote the open ball of $a$ of radius $\epsilon$.

Then:
 * $\ds \forall n \in Z: \map {S_{p^{-n} } } a = \bigcup_{i \mathop = 1}^{p - 1} \map {B_{p^{-n} } } {a + i p^n}$

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

Let $n \in \Z$.

Then:

From Closed Ball is Disjoint Union of Open Balls in P-adic Numbers:
 * $\set {\map {B_{p^{-n} } } {a + i p^n}: i = 0, \dots, p - 1}$ is a set of pairwise disjoint open balls.

Continuing from above:

Also see

 * Closed Ball is Disjoint Union of Open Balls in P-adic Numbers