# 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:

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

## Proof

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

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

$\norm{\paren {x -a} - ip^n}_p \le p^{-m}$

and

$\norm{\paren {x -a} - jp^n}_p \le p^{-m}$

Then:

 $\displaystyle \norm{ip^n - jp^n}_p$ $\le$ $\displaystyle p^{-m}$ Corollary to P-adic Metric on P-adic Numbers is Non-Archimedean Metric $\, \displaystyle \leadsto \,$ $\displaystyle \norm{p^n}_p \norm{i - j}_p$ $\le$ $\displaystyle p^{-m}$ Norm axiom (N2) : (Mulitplicativity) $\, \displaystyle \leadsto \,$ $\displaystyle p^{-n} \norm{i - j}_p$ $\le$ $\displaystyle p^{-m}$ Definition of $p$-adic norm $\, \displaystyle \leadsto \,$ $\displaystyle \norm{i - j}_p$ $\le$ $\displaystyle p^{n-m}$ Multiplying both sides by $p^n$. $\, \displaystyle \leadsto \,$ $\displaystyle p^\paren{m-n}$ $\divides$ $\displaystyle \paren{i - j}$ Definition of $p$-adic norm $\, \displaystyle \leadsto \,$ $\displaystyle j$ $\equiv$ $\displaystyle i \mod p^\paren{m-n}$ Definition of congruence modulo p $\, \displaystyle \leadsto \,$ $\displaystyle i$ $=$ $\displaystyle j$ Integer is Congruent to Unique Integer less than Modulus $\, \displaystyle \leadsto \,$ $\displaystyle \map {B^{\,-}_{p^{-m} } } {a + i p^n}$ $=$ $\displaystyle \map {B^{\,-}_{p^{-m} } } {a + j p^n}$

The result follows.

$\blacksquare$