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:
 * $\forall n \in Z : \set{\map {B^-_{p^{-m} } } {a + i p^n} : i = 0, \dotsc, p^{\paren {m - n}} - 1}$ is a set of pairwise disjoint open balls.

Proof
Let $0 \le i, j \le p^{\paren {m - n}} - 1$.

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

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

and:
 * $\norm {\paren {x -a} - j p^n}_p \le p^{-m}$

We have that P-adic Norm satisfies Non-Archimedean Norm Axioms.

Then:

The result follows.