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:

From set equality:
 * $\map {B^-_{p^{-n} } } a = a + p^n \Z_p$

The result follows.

Also see

 * Open Balls of P-adic Number


 * Local Basis of P-adic Number