Talk:Cauchy sequence

message: Cauchy net redirects here, yet there seems to be nothing about the concept here.... Vivacissamamente
: This comment was left in 2004, it is no longer true. --JBL (talk) 13:06, 13 December 2018 (UTC)
thread_title:  Untitled 

message: How about some sort of formal definition? My elementary analysis textbook states:
A sequence (s_n) 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)
replies: text: 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)

text: 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)
replies: text:  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)
thread_title:  Definition 

message: 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)
replies: text: 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)
thread_title:  p-adic material out of place 

message: 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)
replies: text: 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)
thread_title: 'All two' or 'any two'?

message: 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)
replies: text:  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)
replies: text:  Have put in Spivak reference ( it's even in Wiki in its own right).
Messagetolove 19:15, 26 May 2007 (UTC)
thread_title:  Reference List 

message: 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)
replies: text: 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)
replies: text:  i thank your kindness. :-) --218.42.230.61 01:41, 6 August 2007 (UTC)
thread_title:  question 

message: I would simply like to point out that there is a squiggle on this page that assumes a fair bit of pre-knowledge that is not documented by a re-link into Wikipedia explaining the character symbol and associated concepts
Generalizing, it would be nice if every mathematical symbol used an article had a link to all that the reader should know about it. Could Wikipedia make this some kind of standard?
Maybe this could be further generalized to some kind of overall standard that any relevant concept with sufficient complexity and obscurity be re-linked back into Wikipedia, at least with a requested article.  —Preceding unsigned comment added by Jjalexand (talk • contribs) 12:50, 2 September 2008 (UTC)
thread_title:  A squiggle I dont know how to say or from what alphabet it comes 

message: I think this is not true:
:The values of the exponential, sine and cosine functions, exp(x), sin(x), cos(x), are known to be irrational for any rational value of x≠0, [...]
Counter-example: \cos(60^{\circ}) = \frac{1}{2} \,.
(Francisco Albani (talk) 00:31, 22 September 2008 (UTC)
replies: text: The statement you quoted above is correct in "natural units" which in the case of the trigonometric functions means radians instead of degrees: 60° = 60•2π/360 = π/3, and a rational multiple of an irrational number (π) is still irrational. — Tobias Bergemann (talk) 11:29, 22 September 2008 (UTC)
thread_title:  Values of exp, sin and cos always irrational? 

message: Is there any reason "m,n > N" is better than "m,n >= N"? With ">", it seems to me that the first element is always discarded -- which doesn't matter, I guess, but made me wonder if that is necessary or not. 88.65.186.193 (talk) 14:08, 25 April 2009 (UTC)
replies: text:  In short: it means the same but takes one character less... --CiaPan (talk) 07:50, 1 March 2012 (UTC)
thread_title:  "m,n > N" vs. "m,n >= N" 

message: Why don't we just simply state:
:a Cauchy sequence convergence in a metric space, though the limit might not in the metric space.
Jackzhp (talk) 22:56, 17 June 2009 (UTC)
replies: text: Because when we say something converges to something else, we implicitly suggest that the limit is in the space? I mean, if the limit is not in the space, where is it? It's not easy to talk about things that are outside the space. -- Taku (talk) 21:39, 18 June 2009 (UTC)
replies: text:  That's right. I'd even add some emphasis and say 'because by definition a sequence is convergent if and only if it has a limit,' and a limit of a sequence is defined as an element of the same space to which the sequence's items belong. That's why 'the same' sequence of (1/n)_{n\in\mathbb N} is convergent in real numbers (with a limit equal zero) and is divergent in \mathbb R^+=(0,\infty) --CiaPan (talk) 12:05, 2 July 2009 (UTC)
thread_title: Convergence

message: What about generalization into general topology? I suggest a definition:
(Cn) is Cauchy iff there is a set such that any its open covering contains an open set S such that there exists an integer N such that for any n>N, Cn is in S.
Is it good? Does it appear somewhere?  —Preceding unsigned comment added by 84.229.68.163 (talk) 06:56, 2 July 2009 (UTC)
replies: text: This sounds like the definition of convergence. By definition, a sequence x_n converges to x if every open set containing x contains infinitely many x_n. The problem with generalizing Cauhy sequence is that we need some way to measure the distance between two points; this can be done in a metric space, of course. But in general, not possible. So, we need uniform structure or something else. -- Taku (talk) 11:09, 2 July 2009 (UTC)
replies: text:  'Contains infinitely many' seems too weak for me. Consider the real sequence a_n = \sin n It has no limit in the (0,1) interval although any open subset of (0,1) contains infinitely many terms of (a_n). May be we can say 'any open set containing x contains ALL x_n for n greater than some k,' that is 'all except at most a finite number ot initial terms.' --CiaPan (talk) 07:14, 3 July 2009 (UTC)
replies: text: You're absolutely right. A simpler counterexample would be "1, 0, 1, 0, 1, 0, ...". It should be "x_n -> x if every neighborhood of x contains all x_n except some finite many terms." -- Taku (talk) 13:43, 3 July 2009 (UTC)
thread_title:  General topology 

message: In my experience, the name "Cauchy" is pronounced in English as . Should I add this pronunciation to the article, or wait for someone to come along with a reliable source, or at least say, "yeah, I pronounce it that way, too"? —Tanner Swett (talk) 19:28, 21 June 2011 (UTC)
replies: text: I wouldn't know, although one could in principle search YouTube for evidence that people really pronounce it this way, but I've trusted your observation, been bold and added the pronunciation you are suggesting to the article. This way, it is more visible, and the likelihood of people noticing it and complaining or changing it if it is not the most frequent pronunciation is higher. --Florian Blaschke (talk) 15:46, 6 March 2014 (UTC)
thread_title:  English pronunciation of "Cauchy" 

message: I think there is a problem with the definition of the completion of a topological group G when G does not have a countable basis of open neighbourhood of 0.
In particular, if K is a field with a Krull valuation, whose value group does not have countable cofinality, then, according to the definition given here, K will be complete (since the only Cauchy sequences in K, as defined here, are eventually constant), contrary to usual terminology.
193.206.101.2 (talk) 14:11, 4 July 2013 (UTC)
replies: text: Another example of the same problem: let G be the additive group of the Banach space \ell^1, equipped with the weak topology.  Then every weakly Cauchy SEQUENCE converges in norm, by Schur's property and the fact that L^1-spaces are weakly sequentially complete.  However the group (the Banach space) is certainly not weakly complete.  In order to fix this, one should go to the much more complex setting of uniform spaces and Cauchy filters. Bdmy (talk) 18:32, 4 July 2013 (UTC)
thread_title:  Generalizations to topological groups 

message: the definition "a sequence whose elements become arbitrarily close to each other" I think to be more clear should say "a sequence whose elements become increasingly arbitrarily close to each other", unless a definition of arbitrarily is provided that includes the notion. because, in plain English, the arbitrary number could be 1 and the integers would then be arbitrarily close to one another. Alternatively, could say the sequence is montonically something or other or some other term. 69.201.168.196 (talk) 13:50, 15 July 2014 (UTC)
replies: text: Even though "arbitrarily close" is an expression that (I'm pretty sure) only mathematicians understand, nobody at all will understand "increasingly arbitrarily close". I think the text is acceptable already because the next sentence spells out what "arbitrarily close" means. "Monotonic" is mathematically incorrect: the differences do not have to continually decrease, but can go up and down provided that they eventually get small and stay small. McKay (talk) 05:34, 16 July 2014 (UTC)

text: No, arbitrarily close doesn't mean one could choose distance equal 1 and declare integers are arbitrarily close. Arbitrarily close means whatever distance you choose (whatever small in this case) the sequence terms eventually get closer to each other than your chosen distance; not only they will get closer than some specific threshold you chose (e.g. 1), but they will get closer than any threshold you can choose. --CiaPan (talk) 14:41, 22 July 2014 (UTC)
thread_title:  arbitrarily 

message: Most of the LaTeX in this article is broken.
174.124.71.135 (talk) 21:51, 20 October 2016 (UTC)
thread_title:  Broken LaTeX in this Article 

message: In the 2nd paragraph of the introductory section of the article, the harmonic series is cited as an example of a series that does not converge, and then a formal definition of a Cauchy sequence is presented, intending to show that the harmonic series does not satisfy this definition.
However, the way this formal definition is presented here, the harmonic series in fact does satisfy it! For any given epsilon > 0, simply choose a value of N such that 1/N < epsilon. Then it is clear that for any pair m,n > N, |1/m - 1/n| < epsilon will indeed hold. 
The actual formal definition that is necessary for the Cauchy convergence criterion is correctly stated on the pages for "Convergent series" (section "Cauchy convergence criterion") and "Cauchy's convergence test". In short, it is necessary that for any given epsilon > 0, there exists an N such that the absolute value of *any arbitrarily long sum of terms* beyond the Nth term is less than epsilon.  — Preceding unsigned comment added by Rokirovka (talk • contribs) 11:29, 9 May 2018 (UTC)
replies: text: Are you possibly confusing the harmonic sequence and the harmonic series...? The harmonic sequence is Cauchy, as you proove above, and convergent. However, the harmonic series, which is a sequence of partial sums of the harmonic sequence, is not convergent and not Cauchy, either. It does satisfy an intuitive requirement that neighboring terms are arbitrarily close to one another, but it does not satisfy the definition requirement that almost all terms (which means: all terms from some point onwards) are arbitrarily close to each other. --CiaPan (talk) 11:48, 9 May 2018 (UTC)

text: (Forgot to ping. --CiaPan (talk) 11:50, 9 May 2018 (UTC)
replies: text:   Perhaps the issue here is the precise meaning of the notation. When I read the notation "a_n", whether in reference to the harmonic sequence or to the harmonic series, I interpret that as "the nth term of the harmonic sequence", that is, "1/n". I do not interpret the notation "a_n" as "the nth partial sum 1 + 1/2 + 1/3 + ... + 1/n", even if the preceding material uses the terminology "harmonic series" rather than "harmonic sequence". I think many other readers will have the same instinctive interpretation of this notation as I do, and thus I think the paragraph as it reads right now will be confusing and unclear to many readers. Rokirovka (talk) 15:06, 9 May 2018 (UTC)
replies: text:   But a_n does not refer to a harmonic series or sequence at all! The series is shown as an example for what is not sufficient, then the 'more formal' description shows what is necessary. The intervening sentence starting with 'rather' emphasises that the latter is somewhat opposite to the former, hence the symbols used do not describe the same thing. The a_n symbol is not associated with the \sum\frac 1n series in the text, neither directly nor implicitly. --CiaPan (talk) 15:18, 9 May 2018 (UTC)
replies: text:  I agree with Rokirovka that the paragraph in question is not written very clearly and could lead to confusion (even though it does not contain any unambiguously false statements).  --JBL (talk) 15:20, 9 May 2018 (UTC)
replies: text: Yeah, maybe just using a quick example like the sequence a_n = \sqrt{n} could work just as well without any need to complicate the issue with series?  –Deacon Vorbis (carbon • videos) 15:40, 9 May 2018 (UTC)
thread_title:  Problem with formal definition and harmonic series example 

message: It seems to me that a Cauchy sequence must be an infinite sequence.  Is that correct? If so, I think "infinite" should be part of the definition of Cauchy sequence. There can be finite sequences, but a Cauchy sequence surely must be infiite. Dratman (talk) 21:15, 17 July 2021 (UTC)
thread_title:  Infinite sequence 

message: If the official Wikipedia App is supporting dark mode it's strange to see that some articles prefer this recommended style while most articles do it the other way and it works. So how can we solve this problem without this style change? It is definitely no solution to use light mode here because this should work out of box and my edit was the only workaround I could see from other major math articles. 2A02:6D40:21F1:D200:8542:4B01:AC3B:E45C (talk) 07:25, 12 February 2022 (UTC)
replies: text: This is a known issue (see  and ) and the correct solution is not to break other things (colon-indenting equations apparently causes issues for screen-readers which, unlike dark mode, is not something readers can easily adjust on their end) but to leave comments or ticket requests in the hope the developers of the app fix the issue sooner rather than later. --JBL (talk) 21:28, 12 February 2022 (UTC)
thread_title:  Dark Mode Compatibility in Wikipedia App 

message: if you don't object the information about the name "fundamental sequence", why don't you move it then to the place you want it instead of reverting my edit? That is not a valid excuse to revert (see Wikipedia:Revert only when necessary). The name "fundamental sequence" is a common name particularly by Russian authors in functional analysis.--Tensorproduct (talk) 20:17, 1 September 2023 (UTC)
replies: text: Because of the three options (1) don't include the information, (2) include it where you put it, and (3) include it somewhere else, I am indifferent between (1) and (3), whereas (2) is clearly worse.  You evidently think it is worth including, then be my guest -- in a way that doesn't obviously make the article worse. --JBL (talk) 21:12, 1 September 2023 (UTC)
replies: text: @JayBeeEll Explain how it makes the article worse, that is not an objective statement but your opinion. The name "fundamental sequence" is used in many books and I added one reference.--Tensorproduct (talk) 21:15, 1 September 2023 (UTC)
thread_title:  Fundamental sequence