Talk:Limit inferior and limit superior

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properties of limsup and liminf--for instance, their equality when the limit exists, and inequality when the limit does not, is immediately apparent. It also disposes of the apparent asymmetry in cutting off finitely many consecutive starting terms of the sequence (rather than allowing arbitrary finite subsets to be excluded in some way). A picture comes to mind (at least for me) right away using this version, with two sequences meshed together with two different horizontal asymptotes, one giving rise to the liminf and the other the limsup.

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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. --

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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.

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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

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But considering that the terms were probably introduced quite a while ago (when scientists and mathematicians could still master a handful of Latin and English was not necessarily the primary language of choice), plus the fact that "limit inferior" and "limit superior" sound like really daft English,

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The graphic with the red dashed lines illustrating the limsup and liminf for a wave with decaying amplitude is misleading. The limsup and liminf are numbers, not functions, so they should be straight lines. A better graphic would have a more drastically varying amplitude that would settle out to some

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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

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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

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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

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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

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This seems to indicate that there were posets, where all subsets have suprema and infima, but which were not complete lattices. This is not the case; at least not with what I believe are the ordinary definitions of suprema, infima, and complete lattices. Therefore, I either have missed something,

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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

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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, Knowledge is quite clear in its guidelines about how technical articles should be constructed. I have no real problem with the

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I must say however that your approach to learning and to Knowledge 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

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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

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Baby Rudin defines the (real-valued) limsup and liminf in terms of subsequential limits. That is, the limsup is the supremum of the subsequential limits, and the liminf is the infimum of the subsequential limits. I find this definition very simple to grasp and highly intuitive when determining the

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Knowledge 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

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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

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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,

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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

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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

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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).

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mathematician). Since Knowledge 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. --

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As shown in the updated "Sequences of sets" section, there are two commonly used definitions of the inferior and superior limit with sets. The set-convergence definition seems more compatible with modern methods in analysis (i.e., methods that are generalized easily by filter bases). The

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Perhaps this definition deserves a mention in the "Definition for sequences" section; I actually prefer it over the current truncated sequence definition. Certainly truncated sequences should be included, but I'd prefer them to be second, after the subsequential limits definition.

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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 Knowledge).

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Having a Dutch background, I was taught that the terms liminf and limsup stand for "limes inferior" and "limes superior". The Hazewinkel reference thinks so too, but he's also Dutch so maybe that's not convincing. And I don't really have any other references for it.

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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

1339:. 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. 782:'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. 338:

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.

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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.

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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

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There seems to be a mistake in the last § Specialization for sequences and nets "Note that filter bases are generalizations of nets". Filter base are exactly nets for the partial order "reverse inclusion" (being contained in sthg means bigger than

495: 2707:: Source? It is not obvious to me why the abbreviation for a term with similar but different translations to different languages would be based exactly on the English term. We also use "Limes" in German, and it is the Latin word for the concept. 853:. 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 1267: 2321:

I was attempting to read through the "Properties" section for lim sup and lim inf in the case of real numbers. There is so much discussion and notation in there that it is extremely difficult to read. It could definitely use a cleanup.

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set-inclusion definition seems to be more compatible with old methods in measure theory (e.g., probability). Is there a way to unify them? Is there a reference that does? It's desirable to find a unified definition that maintains the

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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.

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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

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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

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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.

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It is the limit of something more like a step (or staircase) function (with rounded steps, like a rectifier/capacitor pair in an electric circuit), rather than a smooth one, based on the illustrated

2472:(finite or infinite) set has an infimum and a supremum. However, I cannot see how the sentence could be interpreted to cover any such situation. If that was intended, it ought to be clarified. 365:

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

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And my answer referred to the meaning of the abbreviations when it said “usage”. The same notation can abbreviate different things in other languages but the English Knowledge doesn’t care.—

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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.

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of course possible extensions. Since at the place of the sentence only limsups and liminfs of sequences (indexed by positive iintegers) are considered, it is enough to demand that every

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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

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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.

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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.

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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.

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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. :)

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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.

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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. --

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fixed amplitude, and the limsup and liminf lines cutting across equal eventual upper and lower bounds. As is stands, the concept is not illustrated, but obscured.

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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.

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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.

2207:(lim, underlined) is commonly used - perhaps it should be mentioned in this article? My opinion is that it should be mentioned, just as glb is mentioned in the 227:

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.

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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

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would have survived, students would have the necessary set theoretic background and thinking to make filter bases and nets natural concepts to learn in

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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: -->

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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

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is a totally ordered complete lattice with the order topology. Then the left and right sides of the equation are trivially well-defined. Because

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could give useful results on any partial order, with the condition that the limit superior/inferior only exists when that supremum/infimum exists.

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However, the Knowledge reference seems to only mention cluster points of sets. I think what is wanted here is "cluster point of a sequence".

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at least in my ears (wouldn't you change the word order?), it might be worth it if someone with access to source material could investigate.

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Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a

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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.)

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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? --

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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

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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.

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Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

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Look at literally any of the English sources cited here and you’ll see this is he prevailing usage in English. The

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https://web.archive.org/web/20070303150958/http://tt.lamf.uwindsor.ca/314folder/analbookfiles/RintexistLebesgue.pdf

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is on you, as the assertor of a contrary claim, not me, to show that the prevailing usage in English is “limes”.—

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Note that it is important that the definition of a limit omit the point f(2) when evaluating the limit as x-: -->

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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the

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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with

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https://www.simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/

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is not about general relativity, is about a simple concept, it deserves an elementary explanation.

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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You can put elementary material on top without inappropriate threats of deleting good material.--

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after discussing it on the closer's talk page. No further edits should be made to this section.

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before doing mass systematic removals. This message is updated dynamically through the template

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read this article. And then what's the point, that editors feel good about how smart they are?

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I am sorry for my wording, in retrospect I should have quietly edited the article to my liking.

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I agree. The illustration is very confusing. The liminf/limsup needs to be better represented.

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exist for filter bases on sets with complete partial orders, but that the general definition

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More generally, these definitions make sense in any partially ordered set, provided the

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If you found an error with any archives or the URLs themselves, you can fix them with

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is strictly less than sup K, so there must be such a neighborhood of either the form {

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has an open neighborhood that does not intersect it, and therefore does not intersect

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have null intersection, a contradiction. Thus, the hypothetical neighborhood around

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The section "The case of sequences of real numbers" needs an examples subsection.

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and on the relationship between the ordering and the topology, the statement that

2601: 2118:(i.e., that limsup is inf sup and liminf is inf sup) between the two concepts. — 2054: 2022: 1340: 1127: 1085: 1053: 649: 113: 2600:. No special action is required regarding these talk page notices, other than 2234:, I am merely saying that it is commonly used enough to be worth mentioning). 2081:

Suggestion: put the definition in here and do not refer to another web page.

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2. When you omit this point, then it's clear that the limit does not exist. --

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Any relationship between liminf A_n B_n and liminf A_n and liminf B_n?

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say b + \varepsilon and we replace "this lower bound" with b - \varepsilon

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article. (Note: I am not arguing this notation is more common than using

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http://tt.lamf.uwindsor.ca/314folder/analbookfiles/RintexistLebesgue.pdf

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Example of sequence of sup_k and inf_k leading to lim sup and lim inf

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do not have infima. It would seem that limits superior and inferior

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It appears that this is very recently proved in the affirmative.

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or the sense of the sentence should be the same as for the shorter

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Furthermore, unless there are restrictions on both the ordering of

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In order to make your foonote visible, I add <references: -->

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Somewhat long proof that my restriction is sufficient: Suppose

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must be an element of that closure, so this is a contradiction.

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This was not what I said. The question at hand is whether the

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The article did not place any restrictions on the ordering of

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Proposal below is unchallenged, so this request is granted.

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for additional information. I made the following changes:

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I'll wait a bit, and then I will revert this article to

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Update on the question boundedness of prime number gaps

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sets might be a bit put off by these extra concepts.

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More generally, these definitions make sense in any

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For now, I've unified the two definitions using the

1985:. So it actually contains no elements greater than 308:=2, since in any sufficiently small neighbourhod of 112:, a collaborative effort to improve the coverage of 2604:using the archive tool instructions below. Editors 2786: 2766: 2253:They are put on wrong places. Should be swapped. 2226: 2199: 1937: 1783: 1756: 1723: 1665:is the supremum of the closure of some element of 1588: 1476: 1395: 1331: 1294: 1261: 743: 640: 556: 489: 291: 208: 304:Surely the example you give does have limit 2 at 2794:stand for "limit" or "limes" superior/inferior. 2781: 2761: 2221: 2218: 2072:The statement in the current version(5/5/09) is 1542: 1501: 1387: 1371: 1309: 1279: 1224: 1218: 1167: 728: 712: 677: 588: 579: 504: 452: 446: 395: 386: 350:Should we have sigma algebras and lattices here? 2419:I do not understand the idea with the sentence 497:. The trouble is that it's quite possible for 2590:This message was posted before February 2018. 1997:. This contradiction arose by assuming that 1075:Suggest split; inadequate information on nets 8: 1581: 1539: 1533: 1498: 1471: 1432: 1256: 1221: 1212: 1173: 980:Knowledge:Make technical articles accessible 633: 594: 549: 510: 484: 449: 440: 401: 1396:{\displaystyle \sup {\overline {A}}=\sup A} 2855:The following is a closed discussion of a 2560:I have just modified one external link on 58: 2779: 2759: 2216: 2187: 2185: 2109:Sequences of sets: unify two definitions? 1977:}. Furthermore, since it is a subset of 1961:, so it does not intersect the interval { 1924: 1771: 1769: 1757:{\displaystyle d\not \in {\overline {K}}} 1744: 1736: 1710: 1569: 1556: 1546: 1521: 1508: 1490: 1459: 1446: 1436: 1421: 1374: 1369: 1323: 1313: 1307: 1286: 1277: 1244: 1231: 1200: 1187: 1177: 1165: 801:Knowledge:Technical terms and definitions 735: 719: 700: 687: 675: 621: 608: 598: 577: 537: 524: 514: 502: 472: 459: 428: 415: 405: 384: 369:needs to be introduced here somewhere. -- 284: 188: 182: 1332:{\displaystyle \sup {\overline {B}}_{0}} 1049:I have uploaded an image which may help. 173:In the interpretation section, it says: 2535: 1584: 636: 552: 209:{\displaystyle x_{n}<b+\varepsilon } 60: 30: 1993:. This contradicts the definition of 18:Talk:Limit superior and limit inferior 2888:It was proposed in this section that 1144:Changes to definition for filter base 7: 2200:{\displaystyle {\underline {\lim }}} 2180:From what I have seen, the notation 106:This article is within the scope of 49:It is of interest to the following 3038:High-priority mathematics articles 3003:𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 2796:𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 2709:𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 763:, which are primarily the work of 670:i think we should include formula 25: 2995:Limit inferior and limit superior 2991:Limit superior and limit inferior 2898:Limit inferior and limit superior 2890:Limit superior and limit inferior 2564:. Please take a moment to review 2562:Limit superior and limit inferior 2249:Superadditivity and subadditivity 2017:, it is the least upper bound of 2013:. Since it is an upper bound of 126:Knowledge:WikiProject Mathematics 3015:The discussion above is closed. 2880: 2068:suggestion about "cluster point" 778:. Of course, it would be sad if 279:domain). For example, f(x)=x, E= 129:Template:WikiProject Mathematics 93: 83: 62: 31: 2921:to the nom for your input, and 2848:Requested move 27 February 2021 1875:. Suppose the neighborhood of 1784:{\displaystyle {\overline {K}}} 146:This article has been rated as 2171:11:39, 15 September 2009 (UTC) 1989:, which is strictly less than 706: 680: 1: 3011:20:56, 27 February 2021 (UTC) 2410:23:35, 11 November 2011 (UTC) 2385:07:16, 15 December 2011 (UTC) 2317:Properties Section Unreadable 2244:19:20, 12 February 2010 (UTC) 2048:04:19, 29 November 2007 (UTC) 2031:22:44, 28 November 2007 (UTC) 1349:18:00, 28 November 2007 (UTC) 1136:05:05, 26 November 2007 (UTC) 1114:21:29, 25 November 2007 (UTC) 1094:21:04, 25 November 2007 (UTC) 658:01:47, 27 November 2007 (UTC) 312:=2, the function has value 2? 268:20:38, 14 December 2007 (UTC) 243:22:46, 13 December 2007 (UTC) 221:b-\varepsilon for all n : --> 120:and see a list of open tasks. 3033:B-Class mathematics articles 2360:17:43, 5 November 2011 (UTC) 2334:17:38, 21 January 2011 (UTC) 2312:21:59, 13 January 2011 (UTC) 2289:18:25, 1 December 2010 (UTC) 1981:, its supremum is less than 1879:takes the first form. Then 1776: 1749: 1689:. Proof: I shall show that 1685:is the least upper bound of 1551: 1441: 1379: 1318: 1182: 1070:20:06, 29 October 2012 (UTC) 755:Recent shape of this article 603: 564:to exist while some of the B 519: 410: 360:18:30, 13 October 2006 (UTC) 2269:18:38, 6 October 2010 (UTC) 2189: 1028:16:25, 22 August 2007 (UTC) 3054: 2699:07:43, 2 August 2019 (UTC) 2682:07:00, 2 August 2019 (UTC) 2621:(last update: 5 June 2024) 2557:Hello fellow Wikipedians, 2227:{\displaystyle \lim \inf } 2063:21:15, 17 April 2018 (UTC) 1835:less than or equal to sup 1621:is then an upper bound of 1295:{\displaystyle \sup B_{0}} 374:19:37, 14 March 2007 (UTC) 344:20:28, 14 March 2007 (UTC) 325:20:28, 14 March 2007 (UTC) 2957:04:34, 7 March 2021 (UTC) 2923:Happy, Healthy Publishing 2834:19:14, 7 March 2021 (UTC) 2804:19:02, 7 March 2021 (UTC) 2743:18:59, 7 March 2021 (UTC) 2717:10:41, 7 March 2021 (UTC) 2464:On the other hand, there 2157:Examples for real numbers 2151:13:57, 18 June 2009 (UTC) 2130:19:21, 17 June 2009 (UTC) 1043:17:08, 19 July 2011 (UTC) 145: 78: 57: 3017:Please do not modify it. 2979:subst:Requested move/end 2862:Please do not modify it. 2787:{\displaystyle \liminf } 2767:{\displaystyle \limsup } 2658:00:38, 16 May 2017 (UTC) 2524:15:04, 25 May 2013 (UTC) 2507:03:48, 24 May 2013 (UTC) 2482:03:25, 23 May 2013 (UTC) 2415:Unnecessary complicated? 1855:is not a lower bound of 1839:and (if necessary) some 1645:is not a lower bound of 1364:has the order topology, 1012:18:11, 4 June 2007 (UTC) 992:23:29, 3 June 2007 (UTC) 952:22:46, 3 June 2007 (UTC) 942:22:46, 3 June 2007 (UTC) 932:18:32, 3 June 2007 (UTC) 905:22:46, 3 June 2007 (UTC) 867:13:35, 3 June 2007 (UTC) 838:12:31, 3 June 2007 (UTC) 824:12:24, 3 June 2007 (UTC) 813:12:31, 3 June 2007 (UTC) 792:12:13, 3 June 2007 (UTC) 759:I will argue that after 152:project's priority scale 2997:– For consistency with 2689:postpositive adjectives 2553:External links modified 2275:I noticed this too and 2102:23:42, 5 May 2009 (UTC) 1938:{\displaystyle J\cap K} 1625:. Proof: Suppose that 109:WikiProject Mathematics 2788: 2768: 2228: 2201: 1939: 1785: 1758: 1725: 1724:{\displaystyle K\in B} 1657:is strictly less than 1590: 1478: 1397: 1333: 1296: 1263: 1058: 745: 642: 558: 491: 293: 210: 39:This article is rated 2789: 2769: 2229: 2202: 1949:which is a member of 1940: 1786: 1759: 1726: 1591: 1479: 1398: 1334: 1297: 1264: 1056: 746: 643: 559: 492: 294: 292:{\displaystyle \cup } 211: 2999:Infimum and supremum 2778: 2758: 2602:regular verification 2434:exist, such as in a 2391:Alternate definition 2215: 2184: 2176:Notation for lim inf 2001:was not a member of 1923: 1883:is bounded below by 1768: 1735: 1709: 1489: 1420: 1368: 1306: 1276: 1164: 674: 576: 501: 383: 283: 181: 132:mathematics articles 2977:This is template {{ 2592:After February 2018 2526:<references: --> 1945:must have a subset 851:secondary education 2784: 2764: 2646:InternetArchiveBot 2597:InternetArchiveBot 2224: 2197: 2195: 1935: 1919:is a filter base, 1815:} or of the form { 1781: 1754: 1721: 1586: 1585: 1474: 1393: 1329: 1292: 1259: 1059: 859:special relativity 855:general relativity 741: 638: 637: 554: 553: 487: 289: 206: 101:Mathematics portal 45:content assessment 2988: 2987: 2959: 2938: 2910: 2894:renamed and moved 2622: 2497:comment added by 2350:comment added by 2259:comment added by 2188: 2105: 2088:comment added by 1859:. Thus for some 1779: 1752: 1705:. Then for some 1693:is an element of 1629:is an element of 1554: 1444: 1382: 1321: 1185: 606: 522: 413: 270: 258:comment added by 245: 233:comment added by 166: 165: 162: 161: 158: 157: 16:(Redirected from 3045: 2983: 2955: 2951: 2944: 2928: 2908: 2904: 2884: 2883: 2877: 2864: 2793: 2791: 2790: 2785: 2773: 2771: 2770: 2765: 2656: 2647: 2620: 2619: 2598: 2545: 2540: 2509: 2455:complete lattice 2436:complete lattice 2362: 2271: 2233: 2231: 2230: 2225: 2206: 2204: 2203: 2198: 2196: 2104: 2082: 1944: 1942: 1941: 1936: 1847:is greater than 1790: 1788: 1787: 1782: 1780: 1772: 1763: 1761: 1760: 1755: 1753: 1745: 1730: 1728: 1727: 1722: 1697:. Suppose that 1649:, so an element 1595: 1593: 1592: 1587: 1574: 1573: 1561: 1560: 1555: 1547: 1526: 1525: 1513: 1512: 1483: 1481: 1480: 1475: 1464: 1463: 1451: 1450: 1445: 1437: 1402: 1400: 1399: 1394: 1383: 1375: 1338: 1336: 1335: 1330: 1328: 1327: 1322: 1314: 1301: 1299: 1298: 1293: 1291: 1290: 1268: 1266: 1265: 1260: 1249: 1248: 1236: 1235: 1205: 1204: 1192: 1191: 1186: 1178: 761:the recent edits 750: 748: 747: 742: 740: 739: 724: 723: 705: 704: 692: 691: 647: 645: 644: 639: 626: 625: 613: 612: 607: 599: 563: 561: 560: 555: 542: 541: 529: 528: 523: 515: 496: 494: 493: 488: 477: 476: 464: 463: 433: 432: 420: 419: 414: 406: 298: 296: 295: 290: 253: 228: 215: 213: 212: 207: 193: 192: 134: 133: 130: 127: 124: 103: 98: 97: 87: 80: 79: 74: 66: 59: 42: 36: 35: 27: 21: 3053: 3052: 3048: 3047: 3046: 3044: 3043: 3042: 3023: 3022: 3021: 3020: 2984: 2975: 2973: 2949: 2942: 2906: 2881: 2860: 2850: 2776: 2775: 2756: 2755: 2665: 2663:Limit or Limes? 2650: 2645: 2613: 2606:have permission 2596: 2570:this simple FaQ 2555: 2550: 2549: 2548: 2541: 2537: 2492: 2489: 2417: 2402:208.107.152.253 2393: 2374: 2345: 2341: 2319: 2300: 2254: 2251: 2213: 2212: 2182: 2181: 2178: 2159: 2137:discrete metric 2111: 2083: 2070: 2040:Oleg Alexandrov 2009:is a member of 1957:is a subset of 1921: 1920: 1766: 1765: 1733: 1732: 1707: 1706: 1565: 1545: 1517: 1504: 1487: 1486: 1455: 1435: 1418: 1417: 1403:for any subset 1366: 1365: 1312: 1304: 1303: 1282: 1274: 1273: 1240: 1227: 1196: 1176: 1162: 1161: 1146: 1106:Oleg Alexandrov 1077: 1020: 984:Oleg Alexandrov 924:Oleg Alexandrov 830:Oleg Alexandrov 805:Oleg Alexandrov 784:Oleg Alexandrov 757: 731: 715: 696: 683: 672: 671: 668: 617: 597: 574: 573: 567: 533: 513: 499: 498: 468: 455: 424: 404: 381: 380: 352: 281: 280: 276: 216:for all n : --> 184: 179: 178: 171: 131: 128: 125: 122: 121: 99: 92: 72: 43:on Knowledge's 40: 23: 22: 15: 12: 11: 5: 3051: 3049: 3041: 3040: 3035: 3025: 3024: 3014: 2986: 2985: 2974: 2962: 2961: 2943:P.I. Ellsworth 2911: 2902: 2885: 2873: 2872: 2871: 2857:requested move 2851: 2849: 2846: 2845: 2844: 2843: 2842: 2841: 2840: 2839: 2838: 2837: 2836: 2815: 2814: 2813: 2812: 2811: 2810: 2809: 2808: 2807: 2806: 2783: 2763: 2722: 2721: 2720: 2719: 2674:49.198.125.140 2664: 2661: 2640: 2639: 2632: 2585: 2584: 2576:Added archive 2554: 2551: 2547: 2546: 2534: 2533: 2529: 2528: 2527: 2499:67.242.141.205 2488: 2485: 2462: 2461: 2460: 2459: 2443: 2442: 2441: 2440: 2416: 2413: 2392: 2389: 2388: 2387: 2370: 2340: 2337: 2318: 2315: 2299: 2296: 2294: 2292: 2291: 2261:146.50.196.179 2250: 2247: 2223: 2220: 2194: 2191: 2177: 2174: 2158: 2155: 2154: 2153: 2110: 2107: 2069: 2066: 2036: 2035: 2034: 2033: 1934: 1931: 1928: 1899:has the form { 1778: 1775: 1751: 1748: 1743: 1740: 1720: 1717: 1714: 1677: 1676: 1675: 1674: 1616: 1615: 1614: 1605: 1596: 1583: 1580: 1577: 1572: 1568: 1564: 1559: 1553: 1550: 1544: 1541: 1538: 1535: 1532: 1529: 1524: 1520: 1516: 1511: 1507: 1503: 1500: 1497: 1494: 1484: 1473: 1470: 1467: 1462: 1458: 1454: 1449: 1443: 1440: 1434: 1431: 1428: 1425: 1412: 1392: 1389: 1386: 1381: 1378: 1373: 1326: 1320: 1317: 1311: 1289: 1285: 1281: 1270: 1269: 1258: 1255: 1252: 1247: 1243: 1239: 1234: 1230: 1226: 1223: 1220: 1217: 1214: 1211: 1208: 1203: 1199: 1195: 1190: 1184: 1181: 1175: 1172: 1169: 1145: 1142: 1141: 1140: 1139: 1138: 1125: 1117: 1116: 1101: 1100: 1076: 1073: 1051: 1050: 1046: 1045: 1019: 1016: 1015: 1014: 1003: 1002: 1001: 1000: 999: 998: 997: 996: 995: 994: 966: 965: 964: 963: 962: 961: 960: 959: 958: 957: 944: 914: 913: 912: 911: 910: 909: 908: 907: 883: 882: 881: 880: 879: 878: 870: 869: 842: 841: 840: 817: 816: 815: 756: 753: 738: 734: 730: 727: 722: 718: 714: 711: 708: 703: 699: 695: 690: 686: 682: 679: 667: 666:limsup and sum 664: 663: 662: 661: 660: 635: 632: 629: 624: 620: 616: 611: 605: 602: 596: 593: 590: 587: 584: 581: 571: 565: 551: 548: 545: 540: 536: 532: 527: 521: 518: 512: 509: 506: 486: 483: 480: 475: 471: 467: 462: 458: 454: 451: 448: 445: 442: 439: 436: 431: 427: 423: 418: 412: 409: 403: 400: 397: 394: 391: 388: 351: 348: 347: 346: 330: 329: 328: 327: 314: 313: 299:{2}, f:E-: --> 288: 275: 272: 260:24.160.207.229 235:24.160.207.229 224: 223: 218: 205: 202: 199: 196: 191: 187: 170: 169:Interpretation 167: 164: 163: 160: 159: 156: 155: 144: 138: 137: 135: 118:the discussion 105: 104: 88: 76: 75: 67: 55: 54: 48: 37: 24: 14: 13: 10: 9: 6: 4: 3: 2: 3050: 3039: 3036: 3034: 3031: 3030: 3028: 3018: 3013: 3012: 3008: 3004: 3000: 2996: 2992: 2982: 2980: 2972: 2968: 2965: 2960: 2958: 2954: 2952: 2947: 2946: 2945: 2936: 2932: 2927: 2924: 2920: 2916: 2915: 2909: 2901: 2899: 2895: 2891: 2886: 2879: 2878: 2875: 2870: 2868: 2863: 2858: 2853: 2852: 2847: 2835: 2832: 2829: 2825: 2824: 2823: 2822: 2821: 2820: 2819: 2818: 2817: 2816: 2805: 2801: 2797: 2754: 2753:abbreviations 2750: 2746: 2745: 2744: 2741: 2738: 2734: 2730: 2729: 2728: 2727: 2726: 2725: 2724: 2723: 2718: 2714: 2710: 2706: 2702: 2701: 2700: 2697: 2694: 2690: 2686: 2685: 2684: 2683: 2679: 2675: 2669: 2662: 2660: 2659: 2654: 2649: 2648: 2637: 2633: 2630: 2626: 2625: 2624: 2617: 2611: 2607: 2603: 2599: 2593: 2588: 2583: 2579: 2575: 2574: 2573: 2571: 2567: 2563: 2558: 2552: 2544: 2539: 2536: 2532: 2525: 2521: 2517: 2512: 2511: 2510: 2508: 2504: 2500: 2496: 2486: 2484: 2483: 2479: 2475: 2471: 2467: 2458: 2456: 2451: 2450: 2449: 2448: 2447: 2439: 2437: 2433: 2429: 2424: 2423: 2422: 2421: 2420: 2414: 2412: 2411: 2407: 2403: 2397: 2390: 2386: 2382: 2378: 2373: 2369: 2365: 2364: 2363: 2361: 2357: 2353: 2352:66.15.180.216 2349: 2338: 2336: 2335: 2331: 2327: 2323: 2316: 2314: 2313: 2309: 2305: 2297: 2295: 2290: 2286: 2282: 2278: 2274: 2273: 2272: 2270: 2266: 2262: 2258: 2248: 2246: 2245: 2241: 2237: 2210: 2192: 2175: 2173: 2172: 2168: 2164: 2156: 2152: 2148: 2144: 2143: 2138: 2134: 2133: 2132: 2131: 2127: 2123: 2122: 2117: 2108: 2106: 2103: 2099: 2095: 2091: 2087: 2079: 2076: 2073: 2067: 2065: 2064: 2060: 2056: 2050: 2049: 2045: 2041: 2032: 2028: 2024: 2020: 2016: 2012: 2008: 2004: 2000: 1996: 1992: 1988: 1984: 1980: 1976: 1972: 1968: 1964: 1960: 1956: 1952: 1948: 1932: 1929: 1926: 1918: 1914: 1910: 1906: 1902: 1898: 1894: 1890: 1886: 1882: 1878: 1874: 1870: 1866: 1862: 1858: 1854: 1850: 1846: 1842: 1838: 1834: 1830: 1826: 1822: 1818: 1814: 1810: 1806: 1802: 1798: 1794: 1773: 1746: 1741: 1738: 1718: 1715: 1712: 1704: 1700: 1696: 1692: 1688: 1684: 1681: 1680: 1679: 1678: 1672: 1668: 1664: 1660: 1656: 1652: 1648: 1644: 1640: 1636: 1632: 1628: 1624: 1620: 1617: 1613: 1609: 1606: 1604: 1600: 1597: 1578: 1575: 1570: 1566: 1562: 1557: 1548: 1536: 1530: 1527: 1522: 1518: 1514: 1509: 1505: 1495: 1492: 1485: 1468: 1465: 1460: 1456: 1452: 1447: 1438: 1429: 1426: 1423: 1416: 1415: 1413: 1410: 1406: 1390: 1384: 1376: 1363: 1359: 1355: 1354: 1353: 1352: 1351: 1350: 1346: 1342: 1324: 1315: 1287: 1283: 1253: 1250: 1245: 1241: 1237: 1232: 1228: 1215: 1209: 1206: 1201: 1197: 1193: 1188: 1179: 1170: 1160: 1159: 1158: 1156: 1151: 1143: 1137: 1133: 1129: 1123: 1121: 1120: 1119: 1118: 1115: 1111: 1107: 1103: 1102: 1098: 1097: 1096: 1095: 1091: 1087: 1081: 1074: 1072: 1071: 1067: 1063: 1055: 1048: 1047: 1044: 1040: 1036: 1032: 1031: 1030: 1029: 1026: 1017: 1013: 1010: 1005: 1004: 993: 989: 985: 981: 976: 975: 974: 973: 972: 971: 970: 969: 968: 967: 955: 954: 953: 950: 945: 943: 940: 935: 934: 933: 929: 925: 920: 919: 918: 917: 916: 915: 906: 903: 898: 893: 892: 889: 888: 887: 886: 885: 884: 876: 875: 874: 873: 872: 871: 868: 865: 860: 856: 852: 848: 843: 839: 835: 831: 827: 826: 825: 822: 818: 814: 810: 806: 802: 798: 797: 796: 795: 794: 793: 789: 785: 781: 777: 772: 768: 766: 762: 754: 752: 736: 732: 725: 720: 716: 709: 701: 697: 693: 688: 684: 665: 659: 655: 651: 630: 627: 622: 618: 614: 609: 600: 591: 585: 582: 569: 546: 543: 538: 534: 530: 525: 516: 507: 481: 478: 473: 469: 465: 460: 456: 443: 437: 434: 429: 425: 421: 416: 407: 398: 392: 389: 377: 376: 375: 372: 368: 364: 363: 362: 361: 358: 349: 345: 342: 337: 332: 331: 326: 323: 318: 317: 316: 315: 311: 307: 303: 302: 301: 286: 273: 271: 269: 265: 261: 257: 250: 246: 244: 240: 236: 232: 219: 203: 200: 197: 194: 189: 185: 176: 175: 174: 168: 153: 149: 148:High-priority 143: 140: 139: 136: 119: 115: 111: 110: 102: 96: 91: 89: 86: 82: 81: 77: 73:High‑priority 71: 68: 65: 61: 56: 52: 46: 38: 34: 29: 28: 19: 3016: 2989: 2976: 2963: 2940: 2939: 2925: 2912: 2905: 2903: 2887: 2874: 2861: 2854: 2752: 2670: 2666: 2644: 2641: 2616:source check 2595: 2589: 2586: 2559: 2556: 2538: 2530: 2493:— Preceding 2490: 2469: 2465: 2463: 2452: 2444: 2425: 2418: 2398: 2394: 2371: 2367: 2346:— Preceding 2342: 2324: 2320: 2301: 2293: 2252: 2236:Adammanifold 2179: 2160: 2140: 2119: 2115: 2112: 2080: 2077: 2074: 2071: 2051: 2037: 2018: 2014: 2010: 2006: 2002: 1998: 1994: 1990: 1986: 1982: 1978: 1974: 1970: 1966: 1962: 1958: 1954: 1950: 1946: 1916: 1912: 1908: 1904: 1900: 1896: 1892: 1888: 1887:. But then 1884: 1880: 1876: 1872: 1868: 1864: 1860: 1856: 1852: 1848: 1844: 1840: 1836: 1832: 1828: 1824: 1820: 1816: 1812: 1808: 1804: 1800: 1796: 1792: 1702: 1698: 1694: 1690: 1686: 1682: 1670: 1666: 1662: 1658: 1654: 1650: 1646: 1642: 1638: 1634: 1630: 1626: 1622: 1618: 1611: 1607: 1602: 1598: 1408: 1404: 1361: 1357: 1271: 1154: 1149: 1147: 1082: 1078: 1062:Eigenjohnson 1060: 1035:76.66.125.54 1021: 1018:Illustration 896: 776:this version 773: 769: 758: 669: 366: 353: 336:limit points 335: 309: 305: 277: 274:Limit Points 251: 247: 225: 172: 147: 107: 51:WikiProjects 2967:current log 2867:move review 2828:Jasper Deng 2749:Jasper Deng 2737:Jasper Deng 2705:Jasper Deng 2693:Jasper Deng 2339:Bad graphic 2281:Hashproduct 2255:—Preceding 2084:—Preceding 1831:} for some 1791:is closed, 1009:VectorPosse 254:—Preceding 229:—Preceding 123:Mathematics 114:mathematics 70:Mathematics 3027:Categories 2971:target log 2935:page mover 2653:Report bug 2531:References 1915:}. Since 1701:is not in 799:See also 367:definitely 357:Madmath789 2733:WP:BURDEN 2636:this tool 2629:this tool 2470:countable 2326:Coolkid70 2163:LokiClock 2142:TedPavlic 2121:TedPavlic 2090:G. Blaine 1764:. Since 949:TedPavlic 939:TedPavlic 902:TedPavlic 864:TedPavlic 780:TedPavlic 765:TedPavlic 371:TedPavlic 341:TedPavlic 322:TedPavlic 3007:𝗍𝗮𝘭𝙠 2800:𝗍𝗮𝘭𝙠 2713:𝗍𝗮𝘭𝙠 2642:Cheers.— 2516:JoergenB 2495:unsigned 2474:JoergenB 2348:unsigned 2277:fixed it 2257:unsigned 2098:contribs 2086:unsigned 1799:either. 1641:. Then 847:New Math 256:unsigned 231:unsigned 2907:result: 2782:lim inf 2762:lim sup 2566:my edit 2428:suprema 2377:Toolnut 2304:Jackzhp 2209:infimum 2116:duality 1661:. 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