# Closed Balls of P-adic Number

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## 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.

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:

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

## Proof

Let $n \in \Z$.

Then:

\(\displaystyle x \in \map {B^{\,-}_{p^{-n} } } a\) | \(\leadstoandfrom\) | \(\displaystyle \norm {x - a}_p \le p^{-n}\) | Definition of Closed Ball of Normed Division Ring | ||||||||||

\(\displaystyle \) | \(\leadstoandfrom\) | \(\displaystyle p^n \norm {x - a}_p \le 1\) | |||||||||||

\(\displaystyle \) | \(\leadstoandfrom\) | \(\displaystyle \norm {p^{-n} }_p \norm {x - a}_p \le 1\) | Definition of $p$-adic norm | ||||||||||

\(\displaystyle \) | \(\leadstoandfrom\) | \(\displaystyle \norm {p^{-n} \paren {x - a} }_p \le 1\) | Norm axiom (N2) : (Mulitplicativity) | ||||||||||

\(\displaystyle \) | \(\leadstoandfrom\) | \(\displaystyle p^{-n} \paren {x - a} \in \Z_p\) | Definition of $p$-adic integers | ||||||||||

\(\displaystyle \) | \(\leadstoandfrom\) | \(\displaystyle x - a \in p^n \Z_p\) | |||||||||||

\(\displaystyle \) | \(\leadstoandfrom\) | \(\displaystyle x \in a + p^n \Z_p\) |

From set equality:

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

The result follows.

$\blacksquare$