Let a field be equipped with a rank one valuation . A sequence
of elements of is called a zero sequence or a null sequence, if
in the metric induced by .
If together with the metric induced by its valuation is a complete ultrametric field, it’s clear that its sequence (1) has a limit (in ) as soon as the sequence
is a zero sequence.
If is not complete with respect to its valuation , its completion can be made as follows. The Cauchy sequences (1) form an integral domain when the operations “” and “” are defined componentwise. The subset of formed by the zero sequences is a maximal ideal, whence the quotient ring is a field . Moreover, may be isomorphically embedded into and the valuation may be uniquely extended to a valuation of . The field then is complete with respect to and is dense in .