Closed Ball is Disjoint Union of Smaller Closed 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 $a \in \Q_p$.

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


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

Let $n, m \in Z$, such that $n < m$.

Then:
 * $(1) \map {B^{\,-}_{p^{-n}}} a = \displaystyle \bigcup_{i = 0}^{p^{m-n)-1} \map {B^{\,-}_{p^{-m}}} {a + i p^{m-n}}$
 * $(2) \quad\forall n \in Z : \set{\map {B^{\,-}_{p^{-m}}} {a + i p^{m-n}} : i = 0, \dots, p^{m-n}-1}$ is a set of pairwise disjoint closed balls

Also see

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