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

From ProofWiki
Jump to navigation Jump to search

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:

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


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.

$\blacksquare$