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

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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:

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

Proof

Lemma 11

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

$\Box$

Lemma 8

In the real numbers $\R$:

$\ds \lim_{n \mathop \to \infty} \dfrac a {p^{n+1}} = 0$

$\Box$

By definition of convergence:

$\exists n_1 \in \N: \forall n \ge n_1 : - \dfrac 1 2 < \dfrac a {p^{n+1}} < \dfrac 1 2$

Lemma 9

In the real numbers $\R$:

$\ds \lim_{n \mathop \to \infty} \dfrac {a - \paren{p^{n+1} - 1} b } {p^{n+1}} = -b$

$\Box$

By definition of convergence:

$\exists n_2 \in \N: \forall n \ge n_2 : -b - \dfrac 1 2 < \dfrac {a - \paren{p^{n+1} - 1} b } {p^{n+1}} < -b + \dfrac 1 2$

Let:

$n_0 = \max \set{n_1, n_2}$

Then:

$\forall n \ge n_0 : -b - \dfrac 1 2 < \dfrac {a - \paren{p^{n+1} - 1} b } {p^{n+1}} \le r_n \le \dfrac a {p^{n+1}} < \dfrac 1 2$

By hypothesis:

$b \in \Z$ and $\forall n \in \N: r_n \in \Z$

Hence:

$\forall n \ge n_0 : -b \le r_n \le 0$

$\blacksquare$