Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 11

Theorem
Let $p$ be a prime number.

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

Let:
 * $\forall n \in \N: \exists r_n \in \Z : \dfrac a b - \paren{p^{n + 1} \dfrac {r_n} b} \in \set{0, 1, \ldots, p^{n + 1} - 1}$

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

Proof
We have:

The result follows.