Characterisation of Terminal P-adic Expansion

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

Let $x \in \Q_p$.

Then:
 * the $p$-adic expansion of $x$ terminates


 * $\exists a \in \N : \exists k \in \Z : x = \dfrac a {p^k}$