## Theorem

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime number $p$.

Then the image of $\norm {\,\cdot\,}_p$ is:

$\Img {\norm {\,\cdot\,}_p} = \set {p^n : n \in \Z} \cup \set 0$

## Proof

This follows immediately from:

$\blacksquare$