Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 4

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:
 * $\forall n \in \N: \exists A_n, r_n \in \Z$:
 * $(1) \quad \dfrac a b = A_n + p^n \dfrac {r_n} b$
 * $(2) \quad 0 \le A_n \le p^n - 1$

Then:
 * $\forall n \in \N:$
 * $A_{n+1} = A_n + d_n p^n$
 * $r_n = d_n b + p r_{n+1}$

Proof
We have:

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

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

From Integer Coprime to all Factors is Coprime to Whole:
 * $b, p^n$ are coprimes

As $A_{n+1} - A_n \in \Z$:
 * $\dfrac {r_n - p r_{n+1} } b \in \Z$