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 open ball of $a$ of radius $\epsilon$.

Let $n \in Z$.

Let $x, y \in \Q_p$.


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


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


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


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


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