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 a complete 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 $|.|$, 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