# If $(x_n)_{n\geq 1}$, $x_n \in \mathbb{Q}$ is p-adic Cauchy, show ord$_p(x_n)$ eventually constant.

Number Theory 1: Fermat's Dream asks the reader to verify the following. They then use this to extend the definition of Ord$_p$ to $\mathbb{Q}_p$.

Let $p$ be prime and $a\neq 0$.

If $(x_n)_{n\geq 1}$, $n \in \mathbb{N}$ is a p-adic Cauchy sequence of rationals whose class is $a \in \mathbb{Q}_p$, show that ord$_p(x_n)$ is constant for sufficiently large $n$.

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I may be missing something, but I think you require $a\neq 0$ for this to be true. Maybe the problem really says $\mathbb Q_p^\times$? –  Thomas Andrews Jan 31 '13 at 20:03
Correct. I edited that. Thanks. –  user43666 Jan 31 '13 at 20:13
When $a\neq0$, the distance from $x_n$ to $a$ will eventually fall below $|a|_p$. What does the non-archimedean triangle inequality say about $|x_n|_p=|(x_n-a)+a|p$ after that point? –  Jyrki Lahtonen Jan 31 '13 at 21:23