Characterization of Open Ball in P-adic Numbers

Theorem
Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $\Z_p$ be the $p$-adic integers.

For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon} a$ denote the closed ball of center $a$ of radius $\epsilon$.

Let $n \in Z$.

Let $x, y \in \Q_p$.


 * $(1): \quad x \in \map {B_{p^{-n} } } y$


 * $(2): \quad \norm{x - y}_p < p^{-n}$


 * $(3): \quad \map {B_{p^{-n} } } x = \map {B_{p^{-n} } } y$


 * $(4): \quad x - y \in p^{n + 1} \Z_p$


 * $(5): \quad x + p^{n + 1} \Z_p = y + p^{n + 1} \Z_p$

Proof
From P-adic Numbers form Non-Archimedean Valued Field:
 * $\norm {\,\cdot\,}_p$ is a non-Archimedean norm.

Condition $(1)$ iff Condition $(2)$
This follows directly from the definition of a open ball in the $p$-adic numbers.

Condition $(1)$ iff Condition $(3)$
By definition, $\map {B_{p^{-n}}} y$ is an open ball in a non-Archimedean norm $\norm {\,\cdot\,}_p$.

From Centers of Open Balls in Non-Archimedean Division Rings:
 * $x \in \map {B_{p^{-n}}} y \leadsto \map {B_{p^{-n}}} x = \map {B_{p^{-n}}} y$

From Center is Element of Open Ball in P-adic Numbers:
 * $\map {B_{p^{-n}}} x = \map {B_{p^{-n}}} y \leadsto x \in \map {B_{p^{-n}}} x = \map {B_{p^{-n}}} y$

Condition $(3)$ iff Condition $(5)$
From Open Balls of P-adic Number,
 * $\map {B_{p^{-n} } } x = x + p^{n + 1} \Z_p$

and
 * $\map {B_{p^{-n} } } y = y + p^{n + 1} \Z_p$

Hence:
 * $\map {B_{p^{-n} } } x = \map {B_{p^{-n} } } y$ $x + p^{n + 1} \Z_p = y + p^{n + 1} \Z_p$