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.

$\blacksquare$


Also see