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

Proof
From Basis Representation Theorem, $a$ can be expressed uniquely in the form:


 * $\ds a = \sum_{j \mathop = 0}^n d_j p^j$

where:
 * $n$ is such that $p^n \le a < p^{n + 1}$
 * all the $d_j$ are such that $0 \le d_j < p$.

We have:

Let:
 * $m = \begin{cases}

-k & : -k \le 0\\ 0 & : 0 < -k \end{cases}$

For each $i : m \le i \le n-k$, let:
 * $e_i = \begin{cases}

d_{i + k} & : -k \le i \le n-k\\ 0 & : m \le i < -k \end{cases}$

For each $i > n-k$, let:
 * $e_i = 0$

Then:
 * $x = \ds \sum_{i \mathop = m}^\infty e_i p^i$

where:
 * $\forall i \ge m: 0 \le e_i < p$
 * $\forall i > n-k: e_i = 0$

Hence $\ds \sum_{i \mathop = m}^\infty e_i p^i$ is a terminal $p$-adic expansion by definition.

From P-adic Expansion Representative of P-adic Number is Unique, the $p$-adic expansion of $x$ is:
 * $x = \ds \sum_{i \mathop = m}^\infty e_i p^i$