Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 2

Theorem
Let $p$ be a prime number.

Let $a \in \Z, b \in Z_{> 0}$

Let:
 * $\forall n \in \N: \exists r_n \in \Z: \dfrac {a - \paren{p^{n+1} - 1} b } {p^{n+1}} \le r_n \le \dfrac a {p^{n+1}}$

Then:
 * $\exists n_0 \in \N : \forall n \ge n_0 : -b \le r_n \le 0$