User:Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 6

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

Let $x \in \Q_p$.

Let the canonical expansion of $x$ be eventually periodic.

Then:
 * $\exists r \in \Q, n \in \Z, y \in \Z_p$:
 * $(1) \quad x = r + p^n y$
 * $(2) \quad$the canonical expansion of $y$ is periodic

Proof
Let $\ldots d_i \ldots d_2 d_1 d_0. d_{-1} d_{-2} \ldots d_{-m}$ be the canonical expansion of $x$.

By definition of eventually periodic there exists a finite sequence of $k$ digits of $x$:
 * $\tuple {d_{n + k - 1} \ldots d_{n+1} d_n }$

such that $n \ge 0$ and for all $s \in \Z_{\ge 0}$ and for all $j \in \set {0, 2, \ldots, k - 1}$:
 * $d_{n + j + s k} = d_{n + j}$

where $k$ is the smallest $k$ to have this property.

We have:

Let:
 * $\ds r = \begin{cases}

0 & : -m = n\\ \sum_{i \mathop = -m}^{n - 1} d_i p^i & : -m < n \end{cases}$


 * $\ds y = \sum_{i \mathop = n}^\infty d_i p^{i - n}$

Then:
 * $x = r + p^n y$

By definition $r \in \Q$.

Re-indexing the series for $y$, we have:
 * $\ds y = \sum_{i \mathop = 0}^\infty d_{i+n} p^i$

By definition of canonical expansion, the canonical expansion of $y$ is:
 * $\ldots d_{n+i} \ldots d_{n+2} d_{n+1} d_n$

Recall that for all $s \in \Z_{\ge 0}$ and for all $j \in \set {0, 2, \ldots, k - 1}$:
 * $d_{n + j + s k} = d_{n + j}$

By definition of periodic:
 * the canonical expansion of $y$ is periodic