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

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

From Characterization of Open Ball in P-adic Numbers:

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

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.

$\blacksquare$