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:

\(\ds \norm {i p^n - j p^n}_p\)

\(\le\)

\(\ds p^{-m}\)

Corollary to P-adic Metric on P-adic Numbers is Non-Archimedean Metric

\(\ds \leadsto \ \ \)

\(\ds \norm {p^n}_p \norm {i - j}_p\)

\(\le\)

\(\ds p^{-m}\)

Non-Archimedean Norm Axiom $\text N 2$: Multiplicativity

\(\ds \leadsto \ \ \)

\(\ds p^{-n} \norm {i - j}_p\)

\(\le\)

\(\ds p^{-m}\)

Definition of $p$-adic norm

\(\ds \leadsto \ \ \)

\(\ds \norm {i - j}_p\)

\(\le\)

\(\ds p^{n - m}\)

multiplying both sides by $p^n$.

\(\ds \leadsto \ \ \)

\(\ds p^{\paren {m - n} }\)

\(\divides\)

\(\ds \paren {i - j}\)

Definition of $p$-adic norm

\(\ds \leadsto \ \ \)

\(\ds j\)

\(\equiv\)

\(\ds i \mod p^{\paren {m - n} }\)

Definition of Congruence Modulo $p$

\(\ds \leadsto \ \ \)

\(\ds i\)

\(=\)

\(\ds j\)

Integer is Congruent to Integer less than Modulus

\(\ds \leadsto \ \ \)

\(\ds \map {B^-_{p^{-m} } } {a + i p^n}\)

\(=\)

\(\ds \map {B^-_{p^{-m} } } {a + j p^n}\)

The result follows.

$\blacksquare$