Talk:Cauchy sequence
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Cauchy net redirects here, yet there seems to be nothing about the concept here.... Vivacissamamente
- Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that the sequence ought to converge. Nonetheless, Cauchy sequences do not always converge.
Some example please --Taw
added an example, it's kind of kludgy though -- RAE
I'd like to put something along the following lines: Cauchy Seqs are initially useful in spaces such as the Reals because they are a test of convergence which doesn't require a value for the potential limit. -- the flip side is that IF all CSs converge then a space is complete.
There's a sort of switch in perception as things move up a level of abstraction which as a mathematician I find self-evident (and interesting), but I suspect non-mathematicians find baffling or even terrifying:
- theorem: cauchy seqs converge on the Reals
- abstraction: cauchy seqs on other space, where they might not converge
- axiom: part of the defn of complete space
Has this been general idea been coverered anywhere in the maths section? -- Tarquin
I don't think it has been covered; it would fit either here or in complete space. AxelBoldt, Wednesday, June 12, 2002
how about:
- All Cauchy sequences of real or complex numbers converge, hence testing that a sequence is Cauchy is a test of convergence. This is more useful than using the definition of convergence, since that requires the possible limit to be known. With this idea in mind, a metric space in which all Cauchy sequences converge is called complete.
- Thus R and C are complete; but Q is not. The standard construction of the real numbers involves Cauchy sequences of rational numbers; (something about R being the completion of Q...)
...and something on Mathematical abstraction in general somewhere else. I'll see if I can dig up or remember the proof outlines for "Every convergent sequence is a Cauchy sequence" and "every Cauchy sequence is bounded" -- Tarquin June 12 2002
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[edit] Definition
How about some sort of formal definition? My elementary analysis textbook states: A sequence (sn) of real numbers is called a Cauchy sequence if 
66.71.96.78 17:01, 3 October 2005 (UTC)
- The formal definition in the article is more general than yours, applying to metric spaces in general rather than specifically to the real line. —Caesura(t) 17:07, 6 December 2005 (UTC)
- It would also be nice to have Cauchy sequences defined for other absolute values, in particular for p-adic absolute values. Would this be a problem? Gene Ward Smith 09:05, 6 May 2006 (UTC)
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- I guess that could go in the generalization section, as is not really central to the concept of Cauchy sequences as used in analysis. Oleg Alexandrov (talk) 15:48, 6 May 2006 (UTC)
[edit] p-adic material out of place
The p-adic material just after the heading Cauchy sequence in a metric space doesn't seem to belong there. McKay 11:18, 16 June 2006 (UTC)
- I cut it out, together with other fluff. The whole article was a mumbo jumbo of things without clear connections. Oleg Alexandrov (talk) 16:42, 16 June 2006 (UTC)
[edit] 'All two' or 'any two'?
The first paragraph has again been changed to "all two remaining elements ... ". I don't want to start a revert war here, so I would appreciate other views. My view is that it has to be "any two", as "all two" is both mathematically wrong and grammatically wrong. Madmath789 06:38, 21 June 2006 (UTC)
- I agree, but "any two" is not very precise. It is the maximum distance between two of the remaining elements that has to be small. I changed it. McKay 07:57, 21 June 2006 (UTC)
[edit] Reference List
The reference list includes two on algebra and one of constructive mathematics. How about a reference to a good analysis text since after all, Cauchy sequences typically are learned as part of analysis, not algebra.
- Just any analysis text, even if it isn't apparently used as a reference? Why not little Rudin or something? But is that right? There isn't anything, offhand, I can think to add to this article, from a source or otherwise. —vivacissamamente 03:13, 21 October 2006 (UTC)
