User:Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 3

Theorem
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.

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

Let:
 * $\forall n \in \N: \exists A_n, r_n \in \Z$:
 * $(\text a) \quad \dfrac a b = A_n + p^{n+1} \dfrac {r_n} b$
 * $(\text b) \quad \exists n_0 \in \N : \forall n \ge n_0 : -b \le r_n \le 0$

Then:
 * $\ds \lim_{n \mathop \to \infty} A_n = \dfrac a b$

Proof
Let $\epsilon \in \R_{> 0}$.

Let $M = \max \set {\norm{r_0}_p, \norm{r_1}_p, \ldots, \norm{r_{n_0}}_p, \norm{-1}_p, \norm{-2}_p, \ldots, \norm{-b}_p}$

From User:Leigh.Samphier/Sandbox/Power Function on Base Greater than One is Unbounded Above:
 * $\exists N \in \N: p^{N+1} > \dfrac  M {\epsilon \norm b_p}$

We have:

By definition of convergence in $\struct {\Q_p, \norm {\,\cdot\,}_p}$:
 * $\ds \lim_{n \mathop \to \infty} A_n = \dfrac a b$