Local Basis of P-adic Number

Theorem
Let $p$ be a prime number.

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

Let $a \in \Q_p$.

Then the set of open balls $\set {\map {B_{p^{-n} } } a : n \in Z}$ is a local basis of $a$ consisting of clopen sets.

Proof
Let $\BB_a$ be the set of all open balls of $Q_p$ centered on $a$.

That is:
 * $\BB_a = \set{\map {B_\epsilon} a : \epsilon \in \R_{>0}}$

From Open Balls Centered on P-adic Number is Countable:
 * $\BB_a = \set {\map {B_{p^{-n} } } a : n \in Z}$

From Open Balls form Local Basis for Point of Metric Space, $\BB_a$ is a local basis of $a$.

From P-adic Numbers form Non-Archimedean Valued Field:
 * the $p$-adic numbers is a non-Archimedean division ring.

From Open Balls are Clopen In Non-Archimedean Division Ring:
 * the set $\BB_a = \set {\map {B_{p^{-n} } } a : n \in \Z}$ is a local basis of clopen sets.