Limit of non-archimedean absolute values is eventually constant

Question

Let $(x_n)$ be a Cauchy sequence in a field $K$ with respect to a non-archimedean absolute value $|\cdot|$ arising from a discrete valuation $v$ (meaning $|x|:=c^{v(x)}$ for some $c\in(0,1)$. Suppose $\operatorname{lim}|x_n|\neq 0$. I'd like to understand why the sequence $(v(x_n))$ becomes evetually constant.

I found this statement at the top of page three in these notes by Andrew Sutherland.

The only tool at hand seems to be the non-archimedean triangle inequality, but I'm not sure how to use it. What am I missing?

Answer 1

Your questions on $p$-adic numbers are strange, as if to you it was something completely abstract, but $p$-adic numbers are very concrete. If $x\ne 0$ and $x_n\to x$ then $v(x_n-x)>v(x)$ for $n$ large enough so that $v(x_n)=v(x),|x_n|=|x|$. When $K$ is not complete and $(x_n)$ is Cauchy then $x=\lim x_n$ is in its completion instead of $K$ but it doesn't change anything.

$v(x_n-x)>v(x)\implies v(x)=v(x_n)$ is part of the definition:

If $v(x_n) > v(x)$ then $v(x)=v(x_n+(x-x_n))\ge \min(v(x_n),v(x-x_n))$ a contradiction.

Answer 2

When $K$ is a $p$-adic field (rather than just a nonarchimedean field), what you are asking is true, and you have much more tools on hand than just the nonarchimedean inequality. Let $f: K \rightarrow [0,\infty)$ be the function $f(x) = |x|$.

Let $\omega \in K$ be a uniformizer (if $K = \mathbb Q_p$ you can take $\omega = p$), so that $f(\omega) = \rho$ for some $0 < \rho < 1$. Since every $0 \neq x \in K$ is equal to $u \omega^n$ for a unique integer $n$ and unique $u \in K$ with $|u| = 1$, it is clear that

$$\operatorname{Image}(f) = \{0\} \cup \{..., \rho^2, \rho, 1, \rho^{-1}, \rho^{-2}, ... \}$$

Now if $\{x_n\} \in K$ is a Cauchy sequence, then $\{f(x_n)\}$ is a convergent sequence in $\operatorname{Image}(f)$. If we assume that $\{x_n\}$ does not converge to $0$, then $\{f(x_n)\}$ is a convergent sequence in

$$\{..., \rho^2, \rho, 1, \rho^{-1}, \rho^{-2}, ... \}.$$

This last set being discrete (every singleton set is open), the only convergent sequences therein are those which are eventually constant.