Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 5

Theorem
Let $p$ be a prime.

Let $b \in Z_{> 0}$ such that $b, p$ are coprime.

Let $\sequence{d_n}$ be a sequence of $p$-adic digits.

Let $\sequence{r_n}$ be a sequence of integers:
 * $(\text a) \quad \exists n_0 \in \N : \forall n \ge n_0 : -b \le r_n \le 0$
 * $(\text b) \quad \forall n \in \N: r_n = d_n b + p r_{n+1}$

Then:
 * $\exists \mathop m, l \in \N : \forall n \ge m: r_n = r_{n + l}$ and $d_n = d_{n + l}$

Proof
From lemma 3:
 * the sequence $\sequence{r_n}$ takes only finitely many values

Hence:
 * $\exists m, l \in \N : l > 0 : r_m = r_{m+l}$

From lemma 2:

As $b, p$ are coprime:
 * $p \nmid b$

From Divisor of Product:
 * $p \divides \paren {d_m - d_{m + l} }$

By definition of canonical expansion:
 * $d_m, d_{m + l} \in \set{0, 1, \ldots, p-1}$

Hence:
 * $d_m = d_{m+l}$

We have:

Repeating this argument:
 * $\forall n \ge m: r_{n + 1} = r_{n + l + 1}$ and $d_{n + 1} = d_{n + l + 1}$