Characterisation of Terminal P-adic Expansion/Sufficient Condition

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

Let $a \in \N$.

Let $k \in \Z$.

Let $x = \dfrac a {p^k}$.

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