Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 3
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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 Power Function is Unbounded Above:
- $\exists N \in \N: p^{N+1} > \dfrac M {\epsilon \norm b_p}$
We have:
\(\ds \forall n \in \N: n \ge N: \, \) | \(\ds \norm{\dfrac a b - A_n}_p\) | \(=\) | \(\ds \norm{p^{n+1} \dfrac {r_n} b} _p\) | By hypothesis | ||||||||||
\(\ds \) | \(=\) | \(\ds \norm p_p^{n+1} \dfrac {\norm{r_n}_p} {\norm b_p}\) | Norm Axiom $\text N 2$: Multiplicativity | |||||||||||
\(\ds \) | \(\le\) | \(\ds \norm p_p^{n+1} \dfrac M {\norm b_p}\) | Definition of Max Operation | |||||||||||
\(\ds \) | \(\le\) | \(\ds \dfrac 1 {p^{n+1} } \dfrac M {\norm b_p}\) | Definition of P-adic Norm | |||||||||||
\(\ds \) | \(\le\) | \(\ds \dfrac 1 {p^{N+1} } \dfrac M {\norm b_p}\) | Power Function on Base between Zero and One is Strictly Decreasing | |||||||||||
\(\ds \) | \(<\) | \(\ds \dfrac {\epsilon \norm b_p } M \dfrac M {\norm b_p}\) | Choice of $N$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \epsilon\) | Cancelling terms |
By definition of convergence in $\struct {\Q_p, \norm {\,\cdot\,}_p}$:
- $\ds \lim_{n \mathop \to \infty} A_n = \dfrac a b$
$\blacksquare$