Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 3

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

Let $y$ be a rational $p$-adic integer.

Let $\ldots d_n \ldots d_2 d_1 d_0$ be the canonical expansion of $y$.

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

Let:
 * $\forall n \in \N: \exists A_n, r_n \in \Z$:
 * $(\text a) \quad \dfrac a b = A_n + p^{n+1} \dfrac {r_n} b$
 * $(\text b) \quad 0 \le A_n \le p^{n+1} - 1$
 * $(\text c) \quad \dfrac {a - \paren{p^{n+1} - 1} b } {p^{n+1}} \le r_n \le \dfrac a {p^{n+1}}$

Then:
 * $\ds \lim_{n \mathop \to \infty} A_n = \dfrac a b$