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

Contents

[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 \forall\epsilon>0\exists N\mbox{ such that }m,n>N\mbox{ implies }\vert s_n-s_m\vert<\epsilon.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)
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)
I feel that there should be an analysis text in the references:

I suggest M.Spivak's 'Calculus' as an option. It gives a good treatment of real Cauchy sequences, and of the construction of the reals using Cauchy sequences, as well as other constructions (it's definitely an Analysis text, rather than a Calculus text, despite the title). I'll try to look ISBN, etc, if someone doesn't beat me to it. Messagetolove 13:56, 26 May 2007 (UTC)

Have put in Spivak reference ( it's even in Wiki in its own right).

Messagetolove 19:15, 26 May 2007 (UTC)

[edit] Recent edits

Here's my revert. Here are the issues.

  • The sequence
0, 1, 1.5, 2, 2.25, 2.5, 2.75, 3, 3.125, 3.25, ..
appears convergent to me, in spite of what the example claims
  • There is no need to emphasize that
d(xm,xn) < r

implies not only consecutive terms but all remaining terms are getting closer and closer together. It is obvious that we are talking about all terms, since we use different indeces for m and n.

  • Why remove the text about completeness from the def of cauchy sequence? That's the best place to make that point! It is obvious to anybody there that a Cauchy sequence looks as if it is convergent. Then make the reader pay attention right there, rather than moving that text half an article down. Oleg Alexandrov (talk) 15:55, 2 June 2007 (UTC)
Completeness has its own section. Especially with the added issue of counterexamples the two issues should not be mixed up.--Patrick 08:45, 3 June 2007 (UTC)
It looks to me as if Patrick may have intended to be taking the partial sums of the series obtained by adding 1 once, 1/2 2 times, 1/4 4 times, 1/8 8 times, etc that is, essentially the argument used to prove that the harmonic series diverges (shifted a bit). Clearly, if that is the intended sequence, it is not convergent, but it isn't entirely made clear that this is really what is intended, and I haven't heard that referred to as "harmonic" before. I agree with your other points.
Messagetolove 19:06, 2 June 2007 (UTC)
Yes, that is what I mean. However, I wrote that this sequence and the sequence of harmonic numbers are counterexamples, I did not write that this sequence is called harmonic. This sequence has easier numbers than the harmonic numbers, so it is easier to see that it diverges, with all natural numbers occurring in it. If the sequence is not clear enough we can add more terms.--Patrick 22:37, 2 June 2007 (UTC)
But it is much harder to see the pattern in this sequence, even with an explanation. The harmonic series look simpler to me, one just adds 1/n each time. Also, I am not sure this counterexample is relevant here. I'll add a picture these days, that should make things clearer. Oleg Alexandrov (talk) 23:07, 2 June 2007 (UTC)

I reverted the counterexample of consecutive terms, that one is kind of pointless, and poorly written too (it is not the difference of consecutive terms which goes to zero, it is the distance between them, per the previous paragraph). The interesting counterexample is a Cauchy sequence that is not convergent, and that is below. Oleg Alexandrov (talk) 11:44, 3 June 2007 (UTC)

(1) That is not at all an argument for deletion, that is just changing one word.
(2) That is not a counterexample, it is a different (equally interesting) issue.
Patrick 12:15, 3 June 2007 (UTC)
I agree with (1), of course. I don't agree that the thing about consecutive terms is interesting. It is clear enough that the terms are not consecutive from the def. The big deal about Cauchy sequence is relation to completeness. Oleg Alexandrov (talk) 12:24, 3 June 2007 (UTC)

[edit] question

Oleg said

> this is clear enough, n is on the horisontal axis, and x_n is on the vertical one. This is a standard way of graphing functions.

yes, i see this is true. but, the graph ploted by blue points lies on the "plane", not on the axis. so, the blue points do not show the sequence of x_n, do that? sorry for my poor english, thank you. --218.42.230.29 21:48, 5 August 2007 (UTC)

I clarified things a bit saying that what is graphed is not the sequence itself, but its plot. Are things clearer now? Oleg Alexandrov (talk) 00:37, 6 August 2007 (UTC)
i thank your kindness. :-) --218.42.230.61 01:41, 6 August 2007 (UTC)