P-adic Norm Characterisation of Divisibility by Power of p

Theorem
Let $p \in \N$ be a prime.

Let $\Q$ denote the rational numbers.

Let $\norm{\,\cdot\,}$ denote the $p$-adic norm on $\Q$.

Then:
 * $\forall a, b \in \Z: a \equiv b \pmod {p^n} \iff \norm {a - b}_p \le p^{-n}$

Proof
Let $a, b \in \Z$.

We have: