Talk:Limit superior and limit inferior

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[edit] Interpretation

In the interpretation section, it says:

  • The limit superior of xn is the smallest real number b such that, for any positive real number \varepsilon, there exists a natural number N such that x_n<b+\varepsilon for all n > N. In other words, any number larger than the limit superior is an eventual upper bound for the sequence. Only a finite number of elements of the sequence are greater than this upper bound.
  • The limit inferior of xn is the largest real number b that, for any positive real number \varepsilon, there exists a natural number N such that x_n>b-\varepsilon for all n > N. In other words, any number below the limit inferior is an eventual lower bound for the sequence. Only a finite number of elements of the sequence are less than this lower bound.

It seems to me that the last sentence in each of these statements is incorrect. What about the sequence 1/n... it converges to zero, thus the lim sup and lim inf are both zero. However, there are an infinite number of elements greater than 0. I would correct this, but I'm only an undergrad struggling with my first analysis course, so I'm not confident enough about this to change it. Hopefully someone who knows this stuff better can clarify this. —Preceding unsigned comment added by 24.160.207.229 (talk) 22:46, 13 December 2007 (UTC)

Ok... after looking at this again, I see what they meant. The lower/upper bound they are refering to is not the lim inf or the lim sup, it is the number larger than the lim sup or the number smaller than the lim inf. How can this be made clearer? How about instead of saying "this upper bound", we say b + \varepsilon and we replace "this lower bound" with b - \varepsilon

I'm pretty sure this will capture what the original author meant to convey. If I don't hear any objections by Sunday, I'll go ahead and make the change myself. —Preceding unsigned comment added by 24.160.207.229 (talk) 20:38, 14 December 2007 (UTC)

[edit] Limit Points

here is a problem with this definiton: For isolated points of the domain of the function (closure points for the domain of the function, but not limit points) the inferior and superior limits of the function exists while the limit doesn't exist (since the limit is defined only on limit points of the domain). For example, f(x)=x, E=[0,1]\cup{2}, f:E->R. The inferior and superior limits at x=2 both exists and both have value 2 while the limit of the function at x=2 is not defined.

Surely the example you give does have limit 2 at x=2, since in any sufficiently small neighbourhod of x=2, the function has value 2?
Note that it is important that the definition of a limit omit the point f(2) when evaluating the limit as x->2. When you omit this point, then it's clear that the limit does not exist. --TedPavlic 20:28, 14 March 2007 (UTC)
You are correct. The limit inferior and limit superior should be defined in terms of limit points. Limit inferior and limit superior are more general terms that represent the infimum and supremum (respectively) of all limit points of a set. The limit inferior and limit superior of a sequence (or a function) are specializations of this definition. Therefore, the limit inferior, limit superior, and limit all fail to exist at x=2 in the example. So, yes, the article needs to be changed. --TedPavlic 20:28, 14 March 2007 (UTC)

[edit] Should we have sigma algebras and lattices here?

I have a small concern about the section on sequences of sets: since we can define lim sup and lim inf of sequences of sets without ever mentioning sigma algebras and complete lattices, do we really need to mention them here? Just a bit concerned that someone coming here to find out about lim sup of sets might be a bit put off by these extra concepts. Madmath789 18:30, 13 October 2006 (UTC)

It's true that a sequence/function needs to not only be defined over a topological space but a topological space over which the supremum and infimum are defined. That is, a set with order. Order definitely needs to be introduced here somewhere. --TedPavlic 19:37, 14 March 2007 (UTC)
I don't think enough has been done on the order issue. Are limits superior and inferior only defined on complete partial orders? At the moment (now that I have removed some apparently incorrect braces) the article claims that \liminf B = \inf \bigcap \{ \overline{B}_0 : B_0 \in B \} = \sup\{ \inf B_0 : B_0 \in B \}. The trouble is that it's quite possible for \inf \bigcap \{ \overline{B}_0 : B_0 \in B \}\, to exist while some of the B0 do not have infima. It would seem that limits superior and inferior always exist for filter bases on sets with complete partial orders, but that the general definition \liminf B = \inf \bigcap \{ \overline{B}_0 : B_0 \in B \}\, could give useful results on any partial order, with the condition that the limit superior/inferior only exists when that supremum/infimum exists. Dfeuer (talk) 01:47, 27 November 2007 (UTC)

[edit] limsup and sum

i think we should include formula \limsup (a_i + b_i) \le \limsup a_i + \limsup b_i.

[edit] Recent shape of this article

I will argue that after the recent edits, which are primarily the work of TedPavlic, the current article is basically incomprehensible.

Limit superior is already hard enough when people first encounter it, in real analysis, for sequences of real numbers. Starting this article with a fully general partially ordered set, topological spaces, and filters, will assure that nobody who wants to learn about limit superior and inferior will read this article. And then what's the point, that editors feel good about how smart they are?

I'll wait a bit, and then I will revert this article to this version. Of course, it would be sad if TedPavlic's work would go to waste. If this article is reorganized such that the simple definition for sequences of real numbers comes on top, then the definition for sequence of sets, and the more general cases of a filter are treated below, I'd be happy with that too. Oleg Alexandrov (talk) 12:13, 3 June 2007 (UTC)

See also Wikipedia:Technical terms and definitions. Oleg Alexandrov (talk) 12:31, 3 June 2007 (UTC)
You can put elementary material on top without inappropriate threats of deleting good material.--Patrick 12:24, 3 June 2007 (UTC)
Good material in the wrong place is bad material. Of course deleting things is not the best avenue, that's why I'd rather wait for the author of it to organize it. Oleg Alexandrov (talk) 12:31, 3 June 2007 (UTC)
If you review the specialized definition, I think you'll see that its growing complexity makes it far less "simple" than the generalized definition. In fact, I feel the current state of the specialized definition is incomprehensible as it is easily FIVE TIMES LONGER than the generalized definition. I know that the abstraction of filter bases and nets can be difficult to grasp initially; however, their application in higher mathematics makes it important for them to be included here. Additionally, if students would learn the generalized definition first, I think that many other topics in calculus would be trivial. I believe that if New Math would have survived, students would have the necessary set theoretic background and thinking to make filter bases and nets natural concepts to learn in secondary education. All of that being said, while I believe that your original objection may be misguided, I have done as you wish and reorganized the material. Perhaps future versions of this page (and all mathematics pages) must take the same route as the general relativity and special relativity pages that link to an "introduction" page. Or, rather, maybe the common english wiki should be used instead for a similar purpose. Either way, it's sad that you have asked me to highlight material in order to make this page more accessible to the people who need this page the least. Additionally, your threats have discouraged me from taking any more of my time to add material to these pages; did it really have to come to this? --TedPavlic 13:35, 3 June 2007 (UTC)
I am sorry for my wording, in retrospect I should have quietly edited the article to my liking.
I must say however that your approach to learning and to Wikipedia and is completely misguided. I have taught mathematics in universities for eight years, both as an assistant and as a professor. From my experience, people learn in a bottom up approach, starting with simple cases. And this article is not about general relativity, is about a simple concept, it deserves an elementary explanation.
I am unsure that this is a top-vs-bottom discussion. Perhaps it is a discussion about the use of examples. I feel it would be better to use the material on real numbers as an example of a broader subject. That is, I think it would be nice if the material at least hinted that it was a smaller part of a much larger discussion. The previous discussion gave the impression that the infimum and supremum limits should only be applied to real numbers. Such a misconception can lead to difficulties if students proceed into higher mathematics (as a scientist, engineer, or mathematician). Since Wikipedia is an encyclopedia, it would be nice if it were complete. I'm okay with approaching topics with the view that the audience may not be ready for abstraction; however, it's wrong to assume that the audience will always be incapable of the abstract discussion. --TedPavlic 22:46, 3 June 2007 (UTC)
That's why I added a picture to the top of this article. And coming to it after a few months and seeing it completely rewritten from the most abstract and general point of few made me feel very frustrated. But yes, I could have chosen a better approach. Oleg Alexandrov (talk) 18:32, 3 June 2007 (UTC)
I added two sections in front of the previous seven. If your work was completely rewritten, it was not rewritten by me. My additions were the natural extension of the definitions of limsup and liminf on functions of topological spaces. It seemed natural. Perhaps I was wrong. The two small sections that I added make very little dent in the previous seven sections that were there before. I think that's even more clear now that the article has been reorganized. --TedPavlic 22:46, 3 June 2007 (UTC)
Wikipedia is organic and fluid; it is a living document. It's better to have the material available where it can be massaged and reworked later in time. We all seem to agree that all of the material in this article should be preserved. I'm perfectly happy with someone modifying my material. It was not my goal to produce the perfect reference on extremum limits. It was my goal to put more information out there so that other people could work toward that. Additionally, there needs to be more links to nets and filters as that material is pretty awful. I hoped that this article would be a springboard into improving that material. Plus, from my experience with elementary through higher education students, I do believe that replacing all prior mathematics knowledge with a consistent set-theoretic one will benefit all of society greatly. --TedPavlic 22:46, 3 June 2007 (UTC)
Good points. (Note that you did not overwrite anything, it was not my work to start with, it was just a better way to start an article.)
To summarize, articles should start simple (no Bourbaki-like set-theoretic framework, most students are not ready for that). Hinting at more abstract formulations is fine, something like a sentence in the intro could work well. Lastly, a good read is Wikipedia:Make technical articles accessible. Oleg Alexandrov (talk) 23:29, 3 June 2007 (UTC)
If I may insert myself into the discussion, I wish to express my view that Oleg is correct. While reasonable people may differ in their approach to teaching and learning, Wikipedia is quite clear in its guidelines about how technical articles should be constructed. I have no real problem with the compromise article as it stands, but I did remove the part about higher-level math requiring the more technical, generalized definitions. It is just simply not true. There are some applications in higher math that require the general definition, but everyday analysis is full of real-valued functions. The vast majority of people looking at this article will be concerned with learning about limsup and liminf for real-values functions. For those who need more, the article leads them to a higher place at the appropriate point. VectorPosse 18:11, 4 June 2007 (UTC)

[edit] Illustration

I think the illustration is a little confusing. What do the red curves represent? It's easy to mistake them for limsup and liminf, which they aren't, as limsup and liminf are single numbers. I tried to clear up the caption a little. Eighty 16:25, 22 August 2007 (UTC)

[edit] Suggest split; inadequate information on nets

Much argument has gone on about whether this article should start with the general definition for filter bases on ordered topological spaces or a specialized one for sequences. I think the best solution is probably to divide it into two or three articles. One for the reals/sequences case, one for the general case, and probably also one for nets. They can all link to each other and be very friendly. Which one gets the simplest name will of course be up for dispute.

On a slightly different topic: As it stands, the section on nets is entirely inadequate: it is written in a way that suggests the reader should already know the specialized definition for nets, but it neither provides one nor links to one (and I have not found one on Wikipedia). Dfeuer (talk) 21:04, 25 November 2007 (UTC)

I would very much disagree with a split. If you wish, you can create a big article on nets, then, in the current section on nets in this article say that there exists another main article with more info.
There is no need to split the article. The concepts treated here are the same thing, starting from the most particular case to the most general. This is how things should be I believe. Oleg Alexandrov (talk) 21:29, 25 November 2007 (UTC)
That is the way they should be treated didactically. That does not mean they should be so treated in an encyclopedia. It is also not how the section on nets is handled: it is treated only as a particular case of a general principle. I think that needs to be fixed, but I don't understand it well enough to do the fixing, which is why I came here in the first place. Dfeuer (talk) 05:05, 26 November 2007 (UTC)

[edit] Changes to definition for filter base

The article did not place any restrictions on the ordering of X in its definitions of lim sup and lim inf. Without restrictions, these may not exist. I modified the section to acknowledge that they may not exist. I do not know if lim sup and lim inf are generally considered in sets other than complete lattices (in which they are guaranteed to exist). Furthermore, unless there are restrictions on both the ordering of X and on the relationship between the ordering and the topology, the statement that

\sup \bigcap \{ \overline{B}_0 : B_0 \in B \} = \inf\{ \sup B_0 : B_0 \in B \}

will not hold. First, the left hand side may exist, while the right hand side fails to. Second, there may not, in general, be any relationship between \sup B_0 and \sup \overline{B}_0. In the interest of restricting the statement to one I'm fairly confident is true, I added to this claim the restriction that X be a totally ordered complete lattice and have the order topology. A weaker restriction of some sort would probably work as well. Dfeuer (talk) 18:00, 28 November 2007 (UTC)

Somewhat long proof that my restriction is sufficient: Suppose X is a totally ordered complete lattice with the order topology. Then the left and right sides of the equation are trivially well-defined. Because X has the order topology, \sup \overline{A}=\sup A for any subset A of X.
Let
C=\bigcap \{ \overline{B}_0 : B_0 \in B \}
D=\{ \sup B_0 : B_0 \in B \}=\{ \sup \overline{B}_0 : B_0 \in B \}\,
c = sup C
d = inf D
d is then an upper bound of C. Proof: Suppose that x is an element of C and x>d. Then x is not a lower bound of D, so an element y of D is strictly less than x. But y is the supremum of the closure of some element of B, and x must be an element of that closure, so this is a contradiction.
d is the least upper bound of C. Proof: I shall show that d is an element of C. Suppose that d is not in C. Then for some K \in B, d \not \in \overline{K}. Since \overline{K} is closed, d has an open neighborhood that does not intersect it, and therefore does not intersect K either. d is strictly less than sup K, so there must be such a neighborhood of either the form {x:x<b} or of the form {x:a<x<b} for some b less than or equal to sup K and (if necessary) some a. b is greater than d, so b is not a lower bound of D. Thus for some J in B, sup J<b. Suppose the neighborhood of d takes the first form. Then K is bounded below by b. But then J and K have null intersection, a contradiction. Thus, the hypothetical neighborhood around d has the form {x:a<x<b}. Since B is a filter base, J\cap K must have a subset L which is a member of B. L is a subset of K, so it does not intersect the interval {x:a<x<b}. Furthermore, since it is a subset of J, its supremum is less than b. So it actually contains no elements greater than a, which is strictly less than d. This contradicts the definition of d. This contradiction arose by assuming that d was not a member of C, so d is a member of C. Since it is an upper bound of C, it is the least upper bound of C. Dfeuer (talk) 22:44, 28 November 2007 (UTC)

Thanks for your work. My primary concern is that the first part of the article be kept elementary, the way it is now. As far as everything else, there are now upper bounds. :) Oleg Alexandrov (talk) 04:19, 29 November 2007 (UTC)