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
Canonical name	ZeroSequence
Date of creation	2015-07-10 21:03:45
Last modified on	2015-07-10 21:03:45
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	11
Author	pahio (2872)
Entry type	Definition
Classification	msc 40A05
Synonym	null sequence