zero sequence

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

$⟨a_1,a_2,\mathrm{\dots }⟩$(1)

of elements of $k$ is called a zero sequence or a null sequence, if  $\underset{n\to \mathrm{\infty }}{lim}{a}_n=0$ in the metric induced by $|.|$.

If $k$ together with the metric induced by its valuation $|.|$ is acomplete ultrametric field, it’s clear that its sequence(1) has a limit (in $k$) as soon as the sequence

$$⟨a_2-a_1,a_3-a_2,a_4-a_3,\mathrm{\dots }⟩$$

is a zero sequence.

If $k$ is not complete with respect to its valuation $|.|$, itscompletion (http://planetmath.org/Completion) can be made as follows.  TheCauchy sequences (1) form an integral domain $D$ when theoperations “$+$” and “$\cdot$” are defined componentwise.  Thesubset $P$ of $D$ formed by the zero sequences is amaximal ideal, whence the quotient ring $D/P$ is a field$K$.  Moreover, $k$ may be isomorphically embedded into $K$ andthe valuation $|.|$ may be uniquely extended to a valuation of$K$.  The field $K$ then is complete with respect to $|.|$ and $k$is dense in $K$.