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

1328:. 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. 771:'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. 327:

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

484: 2696:: 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. 842:. 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 1256: 2310:

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

2461:(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. 354:

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.

2196:(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 216:

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

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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

1974:. So it actually contains no elements greater than 297:=2, since in any sufficiently small neighbourhod of 101:, a collaborative effort to improve the coverage of 2593:using the archive tool instructions below. Editors 2775: 2755: 2242:They are put on wrong places. Should be swapped. 2215: 2188: 1926: 1772: 1745: 1712: 1654:is the supremum of the closure of some element of 1577: 1465: 1384: 1320: 1283: 1250: 732: 629: 545: 478: 280: 197: 293:Surely the example you give does have limit 2 at 2783:stand for "limit" or "limes" superior/inferior. 2770: 2750: 2210: 2207: 2061:The statement in the current version(5/5/09) is 1531: 1490: 1376: 1360: 1298: 1268: 1213: 1207: 1156: 717: 701: 666: 577: 568: 493: 441: 435: 384: 375: 339:Should we have sigma algebras and lattices here? 2408:I do not understand the idea with the sentence 486:. The trouble is that it's quite possible for 2579:This message was posted before February 2018. 1986:. This contradiction arose by assuming that 1064:Suggest split; inadequate information on nets 8: 1570: 1528: 1522: 1487: 1460: 1421: 1245: 1210: 1201: 1162: 969:Knowledge:Make technical articles accessible 622: 583: 538: 499: 473: 438: 429: 390: 1385:{\displaystyle \sup {\overline {A}}=\sup A} 2844:The following is a closed discussion of a 2549:I have just modified one external link on 47: 2768: 2748: 2205: 2176: 2174: 2098:Sequences of sets: unify two definitions? 1966:}. Furthermore, since it is a subset of 1950:, so it does not intersect the interval { 1913: 1760: 1758: 1746:{\displaystyle d\not \in {\overline {K}}} 1733: 1725: 1699: 1558: 1545: 1535: 1510: 1497: 1479: 1448: 1435: 1425: 1410: 1363: 1358: 1312: 1302: 1296: 1275: 1266: 1233: 1220: 1189: 1176: 1166: 1154: 790:Knowledge:Technical terms and definitions 724: 708: 689: 676: 664: 610: 597: 587: 566: 526: 513: 503: 491: 461: 448: 417: 404: 394: 373: 358:needs to be introduced here somewhere. -- 273: 177: 171: 1321:{\displaystyle \sup {\overline {B}}_{0}} 1038:I have uploaded an image which may help. 162:In the interpretation section, it says: 2524: 1573: 625: 541: 198:{\displaystyle x_{n}<b+\varepsilon } 49: 19: 1982:. This contradicts the definition of 2877:It was proposed in this section that 1133:Changes to definition for filter base 7: 2189:{\displaystyle {\underline {\lim }}} 2169:From what I have seen, the notation 95:This article is within the scope of 38:It is of interest to the following 3027:High-priority mathematics articles 2992:𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 2785:𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 2698:𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 752:, which are primarily the work of 659:i think we should include formula 14: 2984:Limit inferior and limit superior 2980:Limit superior and limit inferior 2887:Limit inferior and limit superior 2879:Limit superior and limit inferior 2553:. Please take a moment to review 2551:Limit superior and limit inferior 2238:Superadditivity and subadditivity 2006:, it is the least upper bound of 2002:. Since it is an upper bound of 115:Knowledge:WikiProject Mathematics 3004:The discussion above is closed. 2869: 2057:suggestion about "cluster point" 767:. Of course, it would be sad if 268:domain). For example, f(x)=x, E= 118:Template:WikiProject Mathematics 82: 72: 51: 20: 2910:to the nom for your input, and 2837:Requested move 27 February 2021 1864:. Suppose the neighborhood of 1773:{\displaystyle {\overline {K}}} 135:This article has been rated as 2160:11:39, 15 September 2009 (UTC) 1978:, which is strictly less than 695: 669: 1: 3000:20:56, 27 February 2021 (UTC) 2399:23:35, 11 November 2011 (UTC) 2374:07:16, 15 December 2011 (UTC) 2306:Properties Section Unreadable 2233:19:20, 12 February 2010 (UTC) 2037:04:19, 29 November 2007 (UTC) 2020:22:44, 28 November 2007 (UTC) 1338:18:00, 28 November 2007 (UTC) 1125:05:05, 26 November 2007 (UTC) 1103:21:29, 25 November 2007 (UTC) 1083:21:04, 25 November 2007 (UTC) 647:01:47, 27 November 2007 (UTC) 301:=2, the function has value 2? 257:20:38, 14 December 2007 (UTC) 232:22:46, 13 December 2007 (UTC) 210:b-\varepsilon for all n : --> 109:and see a list of open tasks. 3022:B-Class mathematics articles 2349:17:43, 5 November 2011 (UTC) 2323:17:38, 21 January 2011 (UTC) 2301:21:59, 13 January 2011 (UTC) 2278:18:25, 1 December 2010 (UTC) 1970:, its supremum is less than 1868:takes the first form. Then 1765: 1738: 1678:. Proof: I shall show that 1674:is the least upper bound of 1540: 1430: 1368: 1307: 1171: 1059:20:06, 29 October 2012 (UTC) 744:Recent shape of this article 592: 553:to exist while some of the B 508: 399: 349:18:30, 13 October 2006 (UTC) 2258:18:38, 6 October 2010 (UTC) 2178: 1017:16:25, 22 August 2007 (UTC) 3043: 2688:07:43, 2 August 2019 (UTC) 2671:07:00, 2 August 2019 (UTC) 2610:(last update: 5 June 2024) 2546:Hello fellow Wikipedians, 2216:{\displaystyle \lim \inf } 2052:21:15, 17 April 2018 (UTC) 1824:less than or equal to sup 1610:is then an upper bound of 1284:{\displaystyle \sup B_{0}} 363:19:37, 14 March 2007 (UTC) 333:20:28, 14 March 2007 (UTC) 314:20:28, 14 March 2007 (UTC) 2946:04:34, 7 March 2021 (UTC) 2912:Happy, Healthy Publishing 2823:19:14, 7 March 2021 (UTC) 2793:19:02, 7 March 2021 (UTC) 2732:18:59, 7 March 2021 (UTC) 2706:10:41, 7 March 2021 (UTC) 2453:On the other hand, there 2146:Examples for real numbers 2140:13:57, 18 June 2009 (UTC) 2119:19:21, 17 June 2009 (UTC) 1032:17:08, 19 July 2011 (UTC) 134: 67: 46: 3006:Please do not modify it. 2968:subst:Requested move/end 2851:Please do not modify it. 2776:{\displaystyle \liminf } 2756:{\displaystyle \limsup } 2647:00:38, 16 May 2017 (UTC) 2513:15:04, 25 May 2013 (UTC) 2496:03:48, 24 May 2013 (UTC) 2471:03:25, 23 May 2013 (UTC) 2404:Unnecessary complicated? 1844:is not a lower bound of 1828:and (if necessary) some 1634:is not a lower bound of 1353:has the order topology, 1001:18:11, 4 June 2007 (UTC) 981:23:29, 3 June 2007 (UTC) 941:22:46, 3 June 2007 (UTC) 931:22:46, 3 June 2007 (UTC) 921:18:32, 3 June 2007 (UTC) 894:22:46, 3 June 2007 (UTC) 856:13:35, 3 June 2007 (UTC) 827:12:31, 3 June 2007 (UTC) 813:12:24, 3 June 2007 (UTC) 802:12:31, 3 June 2007 (UTC) 781:12:13, 3 June 2007 (UTC) 748:I will argue that after 141:project's priority scale 2986:– For consistency with 2678:postpositive adjectives 2542:External links modified 2264:I noticed this too and 2091:23:42, 5 May 2009 (UTC) 1927:{\displaystyle J\cap K} 1614:. Proof: Suppose that 98:WikiProject Mathematics 2777: 2757: 2217: 2190: 1928: 1774: 1747: 1714: 1713:{\displaystyle K\in B} 1646:is strictly less than 1579: 1467: 1386: 1322: 1285: 1252: 1047: 734: 631: 547: 480: 282: 199: 28:This article is rated 2778: 2758: 2218: 2191: 1938:which is a member of 1929: 1775: 1748: 1715: 1580: 1468: 1387: 1323: 1286: 1253: 1045: 735: 632: 548: 481: 283: 281:{\displaystyle \cup } 200: 2988:Infimum and supremum 2767: 2747: 2591:regular verification 2423:exist, such as in a 2380:Alternate definition 2204: 2173: 2165:Notation for lim inf 1990:was not a member of 1912: 1872:is bounded below by 1757: 1724: 1698: 1478: 1409: 1357: 1295: 1265: 1153: 663: 565: 490: 372: 272: 170: 121:mathematics articles 2966:This is template {{ 2581:After February 2018 2515:<references: --> 1934:must have a subset 840:secondary education 2773: 2753: 2635:InternetArchiveBot 2586:InternetArchiveBot 2213: 2186: 2184: 1924: 1908:is a filter base, 1804:} or of the form { 1770: 1743: 1710: 1575: 1574: 1463: 1382: 1318: 1281: 1248: 1048: 848:special relativity 844:general relativity 730: 627: 626: 543: 542: 476: 278: 195: 90:Mathematics portal 34:content assessment 2977: 2976: 2948: 2927: 2899: 2883:renamed and moved 2611: 2486:comment added by 2339:comment added by 2248:comment added by 2177: 2094: 2077:comment added by 1848:. Thus for some 1768: 1741: 1694:. Then for some 1682:is an element of 1618:is an element of 1543: 1433: 1371: 1310: 1174: 595: 511: 402: 259: 247:comment added by 234: 222:comment added by 155: 154: 151: 150: 147: 146: 3034: 2972: 2944: 2940: 2933: 2917: 2897: 2893: 2873: 2872: 2866: 2853: 2782: 2780: 2779: 2774: 2762: 2760: 2759: 2754: 2645: 2636: 2609: 2608: 2587: 2534: 2529: 2498: 2444:complete lattice 2425:complete lattice 2351: 2260: 2222: 2220: 2219: 2214: 2195: 2193: 2192: 2187: 2185: 2093: 2071: 1933: 1931: 1930: 1925: 1836:is greater than 1779: 1777: 1776: 1771: 1769: 1761: 1752: 1750: 1749: 1744: 1742: 1734: 1719: 1717: 1716: 1711: 1686:. Suppose that 1638:, so an element 1584: 1582: 1581: 1576: 1563: 1562: 1550: 1549: 1544: 1536: 1515: 1514: 1502: 1501: 1472: 1470: 1469: 1464: 1453: 1452: 1440: 1439: 1434: 1426: 1391: 1389: 1388: 1383: 1372: 1364: 1327: 1325: 1324: 1319: 1317: 1316: 1311: 1303: 1290: 1288: 1287: 1282: 1280: 1279: 1257: 1255: 1254: 1249: 1238: 1237: 1225: 1224: 1194: 1193: 1181: 1180: 1175: 1167: 750:the recent edits 739: 737: 736: 731: 729: 728: 713: 712: 694: 693: 681: 680: 636: 634: 633: 628: 615: 614: 602: 601: 596: 588: 552: 550: 549: 544: 531: 530: 518: 517: 512: 504: 485: 483: 482: 477: 466: 465: 453: 452: 422: 421: 409: 408: 403: 395: 287: 285: 284: 279: 242: 217: 204: 202: 201: 196: 182: 181: 123: 122: 119: 116: 113: 92: 87: 86: 76: 69: 68: 63: 55: 48: 31: 25: 24: 16: 3042: 3041: 3037: 3036: 3035: 3033: 3032: 3031: 3012: 3011: 3010: 3009: 2973: 2964: 2962: 2938: 2931: 2895: 2870: 2849: 2839: 2765: 2764: 2745: 2744: 2654: 2652:Limit or Limes? 2639: 2634: 2602: 2595:have permission 2585: 2559:this simple FaQ 2544: 2539: 2538: 2537: 2530: 2526: 2481: 2478: 2406: 2391:208.107.152.253 2382: 2363: 2334: 2330: 2308: 2289: 2243: 2240: 2202: 2201: 2171: 2170: 2167: 2148: 2126:discrete metric 2100: 2072: 2059: 2029:Oleg Alexandrov 1998:is a member of 1946:is a subset of 1910: 1909: 1755: 1754: 1722: 1721: 1696: 1695: 1554: 1534: 1506: 1493: 1476: 1475: 1444: 1424: 1407: 1406: 1392:for any subset 1355: 1354: 1301: 1293: 1292: 1271: 1263: 1262: 1229: 1216: 1185: 1165: 1151: 1150: 1135: 1095:Oleg Alexandrov 1066: 1009: 973:Oleg Alexandrov 913:Oleg Alexandrov 819:Oleg Alexandrov 794:Oleg Alexandrov 773:Oleg Alexandrov 746: 720: 704: 685: 672: 661: 660: 657: 606: 586: 563: 562: 556: 522: 502: 488: 487: 457: 444: 413: 393: 370: 369: 341: 270: 269: 265: 205:for all n : --> 173: 168: 167: 160: 120: 117: 114: 111: 110: 88: 81: 61: 32:on Knowledge's 29: 12: 11: 5: 3040: 3038: 3030: 3029: 3024: 3014: 3013: 3003: 2975: 2974: 2963: 2951: 2950: 2932:P.I. Ellsworth 2900: 2891: 2874: 2862: 2861: 2860: 2846:requested move 2840: 2838: 2835: 2834: 2833: 2832: 2831: 2830: 2829: 2828: 2827: 2826: 2825: 2804: 2803: 2802: 2801: 2800: 2799: 2798: 2797: 2796: 2795: 2772: 2752: 2711: 2710: 2709: 2708: 2663:49.198.125.140 2653: 2650: 2629: 2628: 2621: 2574: 2573: 2565:Added archive 2543: 2540: 2536: 2535: 2523: 2522: 2518: 2517: 2516: 2488:67.242.141.205 2477: 2474: 2451: 2450: 2449: 2448: 2432: 2431: 2430: 2429: 2405: 2402: 2381: 2378: 2377: 2376: 2359: 2329: 2326: 2307: 2304: 2288: 2285: 2283: 2281: 2280: 2250:146.50.196.179 2239: 2236: 2212: 2209: 2183: 2180: 2166: 2163: 2147: 2144: 2143: 2142: 2099: 2096: 2058: 2055: 2025: 2024: 2023: 2022: 1923: 1920: 1917: 1888:has the form { 1767: 1764: 1740: 1737: 1732: 1729: 1709: 1706: 1703: 1666: 1665: 1664: 1663: 1605: 1604: 1603: 1594: 1585: 1572: 1569: 1566: 1561: 1557: 1553: 1548: 1542: 1539: 1533: 1530: 1527: 1524: 1521: 1518: 1513: 1509: 1505: 1500: 1496: 1492: 1489: 1486: 1483: 1473: 1462: 1459: 1456: 1451: 1447: 1443: 1438: 1432: 1429: 1423: 1420: 1417: 1414: 1401: 1381: 1378: 1375: 1370: 1367: 1362: 1315: 1309: 1306: 1300: 1278: 1274: 1270: 1259: 1258: 1247: 1244: 1241: 1236: 1232: 1228: 1223: 1219: 1215: 1212: 1209: 1206: 1203: 1200: 1197: 1192: 1188: 1184: 1179: 1173: 1170: 1164: 1161: 1158: 1134: 1131: 1130: 1129: 1128: 1127: 1114: 1106: 1105: 1090: 1089: 1065: 1062: 1040: 1039: 1035: 1034: 1008: 1005: 1004: 1003: 992: 991: 990: 989: 988: 987: 986: 985: 984: 983: 955: 954: 953: 952: 951: 950: 949: 948: 947: 946: 933: 903: 902: 901: 900: 899: 898: 897: 896: 872: 871: 870: 869: 868: 867: 859: 858: 831: 830: 829: 806: 805: 804: 745: 742: 727: 723: 719: 716: 711: 707: 703: 700: 697: 692: 688: 684: 679: 675: 671: 668: 656: 655:limsup and sum 653: 652: 651: 650: 649: 624: 621: 618: 613: 609: 605: 600: 594: 591: 585: 582: 579: 576: 573: 570: 560: 554: 540: 537: 534: 529: 525: 521: 516: 510: 507: 501: 498: 495: 475: 472: 469: 464: 460: 456: 451: 447: 443: 440: 437: 434: 431: 428: 425: 420: 416: 412: 407: 401: 398: 392: 389: 386: 383: 380: 377: 340: 337: 336: 335: 319: 318: 317: 316: 303: 302: 288:{2}, f:E-: --> 277: 264: 261: 249:24.160.207.229 224:24.160.207.229 213: 212: 207: 194: 191: 188: 185: 180: 176: 159: 158:Interpretation 156: 153: 152: 149: 148: 145: 144: 133: 127: 126: 124: 107:the discussion 94: 93: 77: 65: 64: 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 3039: 3028: 3025: 3023: 3020: 3019: 3017: 3007: 3002: 3001: 2997: 2993: 2989: 2985: 2981: 2971: 2969: 2961: 2957: 2954: 2949: 2947: 2943: 2941: 2936: 2935: 2934: 2925: 2921: 2916: 2913: 2909: 2905: 2904: 2898: 2890: 2888: 2884: 2880: 2875: 2868: 2867: 2864: 2859: 2857: 2852: 2847: 2842: 2841: 2836: 2824: 2821: 2818: 2814: 2813: 2812: 2811: 2810: 2809: 2808: 2807: 2806: 2805: 2794: 2790: 2786: 2743: 2742:abbreviations 2739: 2735: 2734: 2733: 2730: 2727: 2723: 2719: 2718: 2717: 2716: 2715: 2714: 2713: 2712: 2707: 2703: 2699: 2695: 2691: 2690: 2689: 2686: 2683: 2679: 2675: 2674: 2673: 2672: 2668: 2664: 2658: 2651: 2649: 2648: 2643: 2638: 2637: 2626: 2622: 2619: 2615: 2614: 2613: 2606: 2600: 2596: 2592: 2588: 2582: 2577: 2572: 2568: 2564: 2563: 2562: 2560: 2556: 2552: 2547: 2541: 2533: 2528: 2525: 2521: 2514: 2510: 2506: 2501: 2500: 2499: 2497: 2493: 2489: 2485: 2475: 2473: 2472: 2468: 2464: 2460: 2456: 2447: 2445: 2440: 2439: 2438: 2437: 2436: 2428: 2426: 2422: 2418: 2413: 2412: 2411: 2410: 2409: 2403: 2401: 2400: 2396: 2392: 2386: 2379: 2375: 2371: 2367: 2362: 2358: 2354: 2353: 2352: 2350: 2346: 2342: 2341:66.15.180.216 2338: 2327: 2325: 2324: 2320: 2316: 2312: 2305: 2303: 2302: 2298: 2294: 2286: 2284: 2279: 2275: 2271: 2267: 2263: 2262: 2261: 2259: 2255: 2251: 2247: 2237: 2235: 2234: 2230: 2226: 2199: 2181: 2164: 2162: 2161: 2157: 2153: 2145: 2141: 2137: 2133: 2132: 2127: 2123: 2122: 2121: 2120: 2116: 2112: 2111: 2106: 2097: 2095: 2092: 2088: 2084: 2080: 2076: 2068: 2065: 2062: 2056: 2054: 2053: 2049: 2045: 2039: 2038: 2034: 2030: 2021: 2017: 2013: 2009: 2005: 2001: 1997: 1993: 1989: 1985: 1981: 1977: 1973: 1969: 1965: 1961: 1957: 1953: 1949: 1945: 1941: 1937: 1921: 1918: 1915: 1907: 1903: 1899: 1895: 1891: 1887: 1883: 1879: 1875: 1871: 1867: 1863: 1859: 1855: 1851: 1847: 1843: 1839: 1835: 1831: 1827: 1823: 1819: 1815: 1811: 1807: 1803: 1799: 1795: 1791: 1787: 1783: 1762: 1735: 1730: 1727: 1707: 1704: 1701: 1693: 1689: 1685: 1681: 1677: 1673: 1670: 1669: 1668: 1667: 1661: 1657: 1653: 1649: 1645: 1641: 1637: 1633: 1629: 1625: 1621: 1617: 1613: 1609: 1606: 1602: 1598: 1595: 1593: 1589: 1586: 1567: 1564: 1559: 1555: 1551: 1546: 1537: 1525: 1519: 1516: 1511: 1507: 1503: 1498: 1494: 1484: 1481: 1474: 1457: 1454: 1449: 1445: 1441: 1436: 1427: 1418: 1415: 1412: 1405: 1404: 1402: 1399: 1395: 1379: 1373: 1365: 1352: 1348: 1344: 1343: 1342: 1341: 1340: 1339: 1335: 1331: 1313: 1304: 1276: 1272: 1242: 1239: 1234: 1230: 1226: 1221: 1217: 1204: 1198: 1195: 1190: 1186: 1182: 1177: 1168: 1159: 1149: 1148: 1147: 1145: 1140: 1132: 1126: 1122: 1118: 1112: 1110: 1109: 1108: 1107: 1104: 1100: 1096: 1092: 1091: 1087: 1086: 1085: 1084: 1080: 1076: 1070: 1063: 1061: 1060: 1056: 1052: 1044: 1037: 1036: 1033: 1029: 1025: 1021: 1020: 1019: 1018: 1015: 1006: 1002: 999: 994: 993: 982: 978: 974: 970: 965: 964: 963: 962: 961: 960: 959: 958: 957: 956: 944: 943: 942: 939: 934: 932: 929: 924: 923: 922: 918: 914: 909: 908: 907: 906: 905: 904: 895: 892: 887: 882: 881: 878: 877: 876: 875: 874: 873: 865: 864: 863: 862: 861: 860: 857: 854: 849: 845: 841: 837: 832: 828: 824: 820: 816: 815: 814: 811: 807: 803: 799: 795: 791: 787: 786: 785: 784: 783: 782: 778: 774: 770: 766: 761: 757: 755: 751: 743: 741: 725: 721: 714: 709: 705: 698: 690: 686: 682: 677: 673: 654: 648: 644: 640: 619: 616: 611: 607: 603: 598: 589: 580: 574: 571: 558: 535: 532: 527: 523: 519: 514: 505: 496: 470: 467: 462: 458: 454: 449: 445: 432: 426: 423: 418: 414: 410: 405: 396: 387: 381: 378: 366: 365: 364: 361: 357: 353: 352: 351: 350: 347: 338: 334: 331: 326: 321: 320: 315: 312: 307: 306: 305: 304: 300: 296: 292: 291: 290: 275: 262: 260: 258: 254: 250: 246: 239: 235: 233: 229: 225: 221: 208: 192: 189: 186: 183: 178: 174: 165: 164: 163: 157: 142: 138: 137:High-priority 132: 129: 128: 125: 108: 104: 100: 99: 91: 85: 80: 78: 75: 71: 70: 66: 62:High‑priority 60: 57: 54: 50: 45: 41: 35: 27: 23: 18: 17: 3005: 2978: 2965: 2952: 2929: 2928: 2914: 2901: 2894: 2892: 2876: 2863: 2850: 2843: 2741: 2659: 2655: 2633: 2630: 2605:source check 2584: 2578: 2575: 2548: 2545: 2527: 2519: 2482:— Preceding 2479: 2458: 2454: 2452: 2441: 2433: 2414: 2407: 2387: 2383: 2360: 2356: 2335:— Preceding 2331: 2313: 2309: 2290: 2282: 2241: 2225:Adammanifold 2168: 2149: 2129: 2108: 2104: 2101: 2069: 2066: 2063: 2060: 2040: 2026: 2007: 2003: 1999: 1995: 1991: 1987: 1983: 1979: 1975: 1971: 1967: 1963: 1959: 1955: 1951: 1947: 1943: 1939: 1935: 1905: 1901: 1897: 1893: 1889: 1885: 1881: 1877: 1876:. But then 1873: 1869: 1865: 1861: 1857: 1853: 1849: 1845: 1841: 1837: 1833: 1829: 1825: 1821: 1817: 1813: 1809: 1805: 1801: 1797: 1793: 1789: 1785: 1781: 1691: 1687: 1683: 1679: 1675: 1671: 1659: 1655: 1651: 1647: 1643: 1639: 1635: 1631: 1627: 1623: 1619: 1615: 1611: 1607: 1600: 1596: 1591: 1587: 1397: 1393: 1350: 1346: 1260: 1143: 1138: 1136: 1071: 1067: 1051:Eigenjohnson 1049: 1024:76.66.125.54 1010: 1007:Illustration 885: 765:this version 762: 758: 747: 658: 355: 342: 325:limit points 324: 298: 294: 266: 263:Limit Points 240: 236: 214: 161: 136: 96: 40:WikiProjects 2956:current log 2856:move review 2817:Jasper Deng 2738:Jasper Deng 2726:Jasper Deng 2694:Jasper Deng 2682:Jasper Deng 2328:Bad graphic 2270:Hashproduct 2244:—Preceding 2073:—Preceding 1820:} for some 1780:is closed, 998:VectorPosse 243:—Preceding 218:—Preceding 112:Mathematics 103:mathematics 59:Mathematics 3016:Categories 2960:target log 2924:page mover 2642:Report bug 2520:References 1904:}. Since 1690:is not in 788:See also 356:definitely 346:Madmath789 2722:WP:BURDEN 2625:this tool 2618:this tool 2459:countable 2315:Coolkid70 2152:LokiClock 2131:TedPavlic 2110:TedPavlic 2079:G. Blaine 1753:. Since 938:TedPavlic 928:TedPavlic 891:TedPavlic 853:TedPavlic 769:TedPavlic 754:TedPavlic 360:TedPavlic 330:TedPavlic 311:TedPavlic 2996:𝗍𝗮𝘭𝙠 2789:𝗍𝗮𝘭𝙠 2702:𝗍𝗮𝘭𝙠 2631:Cheers.— 2505:JoergenB 2484:unsigned 2463:JoergenB 2337:unsigned 2266:fixed it 2246:unsigned 2087:contribs 2075:unsigned 1788:either. 1630:. Then 836:New Math 245:unsigned 220:unsigned 2896:result: 2771:lim inf 2751:lim sup 2555:my edit 2417:suprema 2366:Toolnut 2293:Jackzhp 2198:infimum 2105:duality 1650:. 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