# Closed Ball of P-adic Number

## Theorem

Let $p$ be a prime number.

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

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

Let $a \in \Q_p$.

For all $\epsilon \in \R_{>0}$, let $\map { {B_\epsilon}^-} a$ denote the closed ball of $a$ of radius $\epsilon$.

Then:

$\forall n \in Z : \map {B^-_{p^{-n} } } a = a + p^n \Z_p$

where $a + p^n \Z_p$ denotes the left coset of the principal ideal $p^n \Z_p$ containing $a$ in the subring $\Z_p$.

That is, the closed ball $\map { {B_\epsilon}^-} a$ is the set:

$a + p^n \Z_p = \set{a + p^n z : z \in \Z_p}$

## Proof

Let $n \in \Z$.

Then:

 $\displaystyle x$ $\in$ $\displaystyle \map {B^{\,-}_{p^{-n} } } a$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle \norm {x - a}_p$ $\le$ $\displaystyle p^{-n}$ Definition of Closed Ball in P-adic Numbers $\displaystyle \leadstoandfrom \ \$ $\displaystyle p^n \norm {x - a}_p$ $\le$ $\displaystyle 1$ Multiply both sides of equation by $p^n$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle \norm {p^{-n} }_p \norm {x - a}_p$ $\le$ $\displaystyle 1$ Definition of $p$-adic norm $\displaystyle \leadstoandfrom \ \$ $\displaystyle \norm {p^{-n} \paren {x - a} }_p$ $\le$ $\displaystyle 1$ Norm Axiom $\text N 2$: Multiplicativity $\displaystyle \leadstoandfrom \ \$ $\displaystyle p^{-n} \paren {x - a}$ $\in$ $\displaystyle \Z_p$ Definition of $p$-adic integers $\displaystyle \leadstoandfrom \ \$ $\displaystyle x - a$ $\in$ $\displaystyle p^n \Z_p$ Definition of Principal Ideal $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $\in$ $\displaystyle a + p^n \Z_p$ Definition of Left Coset

From set equality:

$\map {B^-_{p^{-n} } } a = a + p^n \Z_p$

The result follows.

$\blacksquare$