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

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