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

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

and:

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

Then:

 $\displaystyle \norm {i p^n - j p^n}_p$ $\le$ $\displaystyle p^{-m}$ Corollary to P-adic Metric on P-adic Numbers is Non-Archimedean Metric $\displaystyle \leadsto \ \$ $\displaystyle \norm {p^n}_p \norm {i - j}_p$ $\le$ $\displaystyle p^{-m}$ P-adic Norm satisfies Non-Archimedean Norm Axioms: Norm Axiom $\text N 2$: Multiplicativity $\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 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.

$\blacksquare$