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

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

Let $n, m \in Z$, such that $n < m$.


Then:

$(1) \quad\map {B^{\,-}_{p^{-n}}} a = \displaystyle \bigcup_{i = 0}^{p^\paren{m-n}-1} \map {B^{\,-}_{p^{-m}}} {a + i p^n}$
$(2) \quad\set{\map {B^{\,-}_{p^{-m}}} {a + i p^n} : i = 0, \dots, p^\paren{m-n}-1}$ is a set of pairwise disjoint closed balls

Proof

Condition $(1)$

Lemma

$\forall y \in \Q_p: \norm{y}_p \le p^{-n}$ if and only if there exists $i \in \Z$:
$(1)\quad0 \le i \le p^\paren{m-n}-1$
$(2)\quad\norm{y - i p^n}_p \le p^{-m}$


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

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

By definition of a closed ball:

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

From Lemma:

$\norm{ x - a}_p \le p^{-n}$.

By definition of a closed ball:

$x \in \map {B^{\,-}_{p^{-n}}} a$

Since $x$ was arbitrary:

$\map {B^{\,-}_{p^{-m}}} {a + i p^{-n}} \subseteq \map {B^{\,-}_{p^{-n}}} a$

Since $i$ was arbitrary:

$\displaystyle \bigcup_{i = 0}^{p^\paren{m-n}-1} \map {B^{\,-}_{p^{-m}}} {a + i p^{-n}} \subseteq \map {B^{\,-}_{p^{-n}}} a$


Let $x \in \map {B^{\,-}_{p^{-n}}} a$.

By definition of a closed ball:

$\norm{ x - a}_p \le p^{-n}$.

From Lemma:

$\exists i \in \N : 0 \le i \le p^\paren{m-n}-1 :\norm{ x - a - i p^{-n}} \le p^{-m}$

By definition of a closed ball:

$\exists i \in \N : 0 \le i \le p^\paren{m-n}-1 : x \in \map {B^{\,-}_{p^{-m}}} {a + i p^{-n}}$

Hence:

$\map {B^{\,-}_{p^{-n}}} a \subseteq \displaystyle \bigcup_{i = 0}^{p^\paren{m-n}-1} \map {B^{\,-}_{p^{-m}}} {a + i p^{-n}}$


It follows that:

$\map {B^{\,-}_{p^{-n}}} a = \displaystyle \bigcup_{i = 0}^{p^\paren{m-n}-1} \map {B^{\,-}_{p^{-m}}} {a + i p^{-n}}$

$\Box$

Condition $(2)$

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.

$\blacksquare$