Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 10

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

Let:
 * $n, k \in \N : k > 0 : r_n = r_{n + k}$

Then:
 * $d_{n+1} = d_{n + k + 1}$
 * $r_{n+1} = r_{n + k + 1}$

Proof
We have:

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

From Euclid's Lemma:
 * $p \divides \paren {d_{n + 1} - d_{n + k + 1} }$

By definition of $p$-adic digits:
 * $d_{n + 1}, d_{n + k + 1} \in \set{0, 1, \ldots, p-1}$

Hence:
 * $d_{n + 1} = d_{n + k + 1}$

We have: