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

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