Canonical P-adic Expansion of Rational is Eventually Periodic/Necessary Condition

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

Let $x$ be a rational number.

Then:
 * the canonical expansion of $x$ is eventually periodic.

Proof
Let $\ldots d_n \ldots d_2 d_1 d_0. d_{-1} d_{-2} \ldots d_{-m}$ be the canonical expansion of $x$.

It is sufficient to show that the canonical expansion $\ldots d_n \ldots d_2 d_1 d_0$ is eventually periodic.

Let $y$ be the $p$-adic number with canonical expansion:
 * $\ldots d_n \ldots d_2 d_1 d_0$

We have:
 * $y = x - \ds \sum_{i \mathop = -m}^{-1} d_i p^i$

So:
 * $y$ is a rational number

By definition of $p$-adic integer:
 * $y$ is a $p$-adic integer

Let:
 * $y = \dfrac a b : a \in \Z, b \in Z_{> 0}$ are coprime

From Characterization of Rational P-adic Integer:
 * $p \nmid b$

From Prime not Divisor implies Coprime:
 * $b, p$ are coprime

Lemma 2
For all $n \in \N$, let:
 * $A_n = \dfrac a b - \paren{p^{n + 1} \dfrac {r_n} b}$

Then:
 * $\dfrac a b = A_n + \paren{p^{n + 1} \dfrac {r_n} b}$

From Lemma 1, for all $n \in \N$:
 * $A_n \in \N$
 * $0 \le A_n \le p^{n+1} - 1$

Lemma 5
It follows that $\ldots d_n \ldots d_2 d_1 d_0$ is eventually periodic.