# 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:

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

## Proof

$\blacksquare$