Characterisation of Terminal P-adic Expansion/Necessary Condition

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

Let $x \in \Q_p$.

Let the $p$-adic expansion of $x$ terminate.

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

Proof
Let the $p$-adic expansion of $x$ be:
 * $x = \ds \sum_{n \mathop = m}^\infty d_n p^n$

where:
 * $m \in \Z_{\le 0}$
 * $\forall n \in \Z_{\ge m}: d_n$ is a $p$-adic digit
 * $m < 0 \implies d_m \ne 0$

By the definition of terminates:
 * $\exists n_0 \in \N : n_0 \ge m : \forall n \ge n_0 : d_n = 0$

Hence:

Let:
 * $k = -m$

Let:
 * $a = \ds \sum_{n \mathop = 0}^{n_0 - m} d_{n + m} p^n $

Then:
 * $x = \dfrac a {p^k}$

The result follows.