Characterization of Closed 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 $x, y \in \Q_p$.

Let $n \in \Z$.


 * $(1)\quad x \in \map {B^{\,-}_{p^{-n}}} y$


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


 * $(3)\quad \map {B^{\,-}_{p^{-n}}} x = \map {B^{\,-}_{p^{-n}}} y$


 * $(4)\quad x - y \in p^n \Z_p$


 * $(5)\quad x + p^n \Z_p = y + p^n \Z_p$

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 closed ball in the $p$-adic numbers.

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

From Centers of Closed 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 Closed 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 Closed Balls of P-adic Number,
 * $\map {B^{\,-}_{p^{-n}}} x = x + p^n \Z_p$

and
 * $\map {B^{\,-}_{p^{-n}}} y = y + p^n \Z_p$

Hence:
 * $\map {B^{\,-}_{p^{-n}}} x = \map {B^{\,-}_{p^{-n}}} y$ $x + p^n \Z_p = y + p^n \Z_p$