Let $k$ be a field and $|\cdot|$ its non-archimedean valuation of rank one. Then its value group $|k\!\smallsetminus\!\{0\}|$ may be considered to be a subgroup of the multiplicative group of $\mathbb{R}$ . In the completion $K$ of the valued field $k$ , the extension of the valuation is defined by $$|x| := \lim_{n\to\infty}|x_n|,$$ when the Cauchy sequence $x_1,\,x_2,\,\ldots,\,x_n,\,\ldots$ of elements of $k$ determines the element $x$ of $K$ .

Proof. Of course, $|k| \subseteq |K|$ . Let $x = \lim_{n\to\infty}x_n$ be any non-zero element of $K$ , where $x_j$ 's form a Cauchy sequence in $k$ . Then there exists a positive number $n_0$ such that $$|x_n-x| < |x|$$ for all $n > n_0$ . For all these values of $n$ we have $$|x_n| = |x+(x_n-x)| = |x|$$ according to the ultrametric triangle inequality. Thus we see that $|K|\subseteq |k|$ .