# zero sequence

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

 $\displaystyle\langle a_{1},\,a_{2},\,\ldots\rangle$ (1)

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

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 (http://planetmath.org/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 $P$ of $D$ formed by the zero sequences is a maximal ideal, whence the quotient ring $D/P$ 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 $K$.

Title zero sequence ZeroSequence 2015-07-10 21:03:45 2015-07-10 21:03:45 pahio (2872) pahio (2872) 11 pahio (2872) Definition msc 40A05 null sequence