From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Axioms for complex numbers
Date: 20 Nov 1994 04:30:03 GMT

In article <3amckb$10m8@b.stat.purdue.edu>,
Herman Rubin <hrubin@b.stat.purdue.edu> wrote:
>
>In a non-Archimedean ordered field, you will have much more difficulty
>finding Cauchy sequences than is apparent.  The sequence whose n-th
>term is 1/n does not converge.
>
>Cauchy sequences are only useful if 1/n -> 0, which is Archimedean.

Well, useful is as useful does, I suppose. I find lots of useful Cauchy
sequences in the (non-archimedean) complete fields with a finite
residue field, namely, the finite extensions of the quotient fields
F_p((x)) of the power series rings F_p[[x]] over the finite fields F_p.
as well as the P-adics (completions of an algebraic number field
with respect to a prime  P  in its ring of integers). For example,
I've always been partial to the Cauchy sequence 1, 3, 7, 15, ...
in the 2-adics (It's the one that arises while observing the
geometric series \Sum 2^k with ratio r=2 does indeed converge to 
1/(1-r) = -1 ...)

dave