Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Disjoint Closed Balls

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 open $\epsilon$-ball of $a$.

Then:
 * $\forall n \in Z : \set{\map {B_{p^{-n}}} {a + i p^n} : i = 0, \dots, p-1}$ is a set of pairwise disjoint open balls

Proof
Let $0 \le i,j \le p-1$.

Let $x \in \map {B_{p^{-n}}} {a + i p^n} \cap \map {B_{p^{-n}}} {a + j p^n}$

From Characterization of Open Ball in P-adic Numbers:
 * $\norm{\paren {x -a} - ip^n}_p < p^{-n}$

and
 * $\norm{\paren {x -a} - jp^n}_p < p^{-n}$

Then:

The result follows.