Let a field $k$ be equipped with a rank one valuation $|.|$ . A sequence

(1)

If $k$ together with the metric induced by its valuation $|.|$ is a complete ultrametric field, it's clear that its sequence (1) has a limit (in $k$ ) as soon as the sequence $$\langle a_2\!-a_1,\, a_3\!-\!a_2,\,a_4\!-\!a_3,\,\ldots \rangle$$ is a zero sequence.

If $k$ is not complete with respect to its valuation $|.|$ , its completion can be made as follows. The Cauchy sequences (1) form an integral domain $D$ when the operations ``$+$ '' and ``$\cdot$ '' are defined componentwise. The subset $\mathfrak{p}$ of $D$ formed by the zero sequences is a maximal ideal, whence the quotient ring $D/\mathfrak{p}$ is a field $K$ . Moreover, $k$ may be isomorphically embedded into $K$ and the valuation $|.|$ may be uniquely extended to a valuation of $K$ . The field $K$ then is complete with respect to $|.|$ and $k$ is dense in $K$ .