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 $\Z_p$ be the $p$-adic integers.

Let $a \in \Q_p$.

Then the set $\set {a + p^n \Z_p: n \in Z}$ is a local basis of $a$ consisting of clopen sets.

Proof
Let $\sequence{x_n}$ be the real sequence such that:
 * $\forall n \in N : x_n = p^{-n}$

From Sequence of Powers of Number less than One, $\sequence{p^{-n}}$ is a real null sequence.

From Null Sequence induces Local Basis in Metric Space, the set $\set { \map {B_{p^{-n}}} a : n \in \Z}$ is a local basis.

Recall that the $p$-adic norm on the $p$-adic numbers is a non-Archimedean norm by definition.

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

From Open Balls of P-adic Number:
 * $\set { \map {B_{p^{-n}}} a : n \in \Z} = \set {a + p^{n+1} \Z_p : n \in \Z} = \set {a + p^n \Z_p : n \in \Z}$

The result follows.

Also see

 * Closed Balls of P-adic Number


 * Open Balls of P-adic Number