Talk:Limit inferior and limit superior

message: In the interpretation section, it says:
* The limit superior of xn is the smallest real number b such that, for any positive real number \varepsilon, there exists a natural number N such that x_n N. In other words, any number larger than the limit superior is an eventual upper bound for the sequence. Only a finite number of elements of the sequence are greater than this upper bound.
* The limit inferior of xn is the largest real number b that, for any positive real number \varepsilon, there exists a natural number N such that x_n>b-\varepsilon for all n > N. In other words, any number below the limit inferior is an eventual lower bound for the sequence. Only a finite number of elements of the sequence are less than this lower bound.
It seems to me that the last sentence in each of these statements is incorrect.  What about the sequence 1/n... it converges to zero, thus the lim sup and lim inf are both zero.  However, there are an infinite number of elements greater than 0.  I would correct this, but I'm only an undergrad struggling with my first analysis course, so I'm not confident enough about this to change it.  Hopefully someone who knows this stuff better can clarify this.  —Preceding unsigned comment added by 24.160.207.229 (talk) 22:46, 13 December 2007 (UTC)
thread_title:  Interpretation 

message: here is a problem with this definiton: 
For isolated points of the domain of the function (closure points for the domain of the function, but not limit points) the inferior and superior limits of the function exists while the limit doesn't exist (since the limit is defined only on limit points of the domain).
For example, f(x)=x, E=[0,1]\cup{2}, f:E->R. The inferior and superior limits at x=2 both exists and both have value 2 while the limit of the function at x=2 is not defined.
:Surely the example you give does have limit 2 at x=2, since in any sufficiently small neighbourhod of x=2, the function has value 2?
::Note that it is important that the definition of a limit omit the point f(2) when evaluating the limit as x->2. When you omit this point, then it's clear that the limit does not exist. --TedPavlic 20:28, 14 March 2007 (UTC)
replies: text: You are correct. The limit inferior and limit superior should be defined in terms of limit points. Limit inferior and limit superior are more general terms that represent the infimum and supremum (respectively) of all limit points of a set. The limit inferior and limit superior of a sequence (or a function) are specializations of this definition. Therefore, the limit inferior, limit superior, and limit all fail to exist at x=2 in the example. So, yes, the article needs to be changed. --TedPavlic 20:28, 14 March 2007 (UTC)
thread_title:  Limit Points 

message: I have a small concern about the section on sequences of sets: since we can define lim sup and lim inf of sequences of sets without ever mentioning sigma algebras and complete lattices, do we really need to mention them here? Just a bit concerned that someone coming here to find out about lim sup of sets might be a bit put off by these extra concepts. Madmath789 18:30, 13 October 2006 (UTC)
replies: text: It's true that a sequence/function needs to not only be defined over a topological space but a topological space over which the supremum and infimum are defined. That is, a set with order. Order definitely needs to be introduced here somewhere. --TedPavlic 19:37, 14 March 2007 (UTC)
replies: text:  I don't think enough has been done on the order issue.  Are limits superior and inferior only defined on complete partial orders?  At the moment (now that I have removed some apparently incorrect braces) the article claims that \liminf B = \inf \bigcap \{ \overline{B}_0 : B_0 \in B \} = \sup\{ \inf B_0 : B_0 \in B \}.  The trouble is that it's quite possible for \inf \bigcap \{ \overline{B}_0 : B_0 \in B \}\, to exist while some of the B0 do not have infima.  It would seem that limits superior and inferior always exist for filter bases on sets with complete partial orders, but that the general definition \liminf B = \inf \bigcap \{ \overline{B}_0 : B_0 \in B \}\, could give useful results on any partial order, with the condition that the limit superior/inferior only exists when that supremum/infimum exists. Dfeuer (talk) 01:47, 27 November 2007 (UTC)
thread_title:  Should we have sigma algebras and lattices here? 

message: I think the illustration is a little confusing. What do the red curves represent? It's easy to mistake them for limsup and liminf, which they aren't, as limsup and liminf are single numbers. I tried to clear up the caption a little. Eighty 16:25, 22 August 2007 (UTC)
replies: text: I agree. The illustration is very confusing. The liminf/limsup needs to be better represented.76.66.125.54 (talk) 17:08, 19 July 2011 (UTC)

text: I have uploaded an image which may help.
Eigenjohnson (talk) 20:06, 29 October 2012 (UTC)
thread_title:  Illustration 

message: The statement in the current version(5/5/09) is
"The liminf and limsup of a sequence are respectively the smallest and greatest cluster points "
However, the Wikipedia reference  seems to only mention cluster points of sets. I think what is wanted here is "cluster point of a sequence". 
Suggestion: put the definition in here and do not refer to another web page.  —Preceding unsigned comment added by G. Blaine (talk • contribs) 23:42, 5 May 2009 (UTC)
thread_title:  suggestion about "cluster point" 

message: 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 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 duality (i.e., that limsup is inf sup and liminf is inf sup) between the two concepts. —TedPavlic (talk) 19:21, 17 June 2009 (UTC)
replies: text: For now, I've unified the two definitions using the discrete metric. —TedPavlic (talk) 13:57, 18 June 2009 (UTC)
thread_title:  Sequences of sets: unify two definitions? 

message: The section "The case of sequences of real numbers" needs an examples subsection. LokiClock (talk) 11:39, 15 September 2009 (UTC)
thread_title:  Examples for real numbers 

message: From what I have seen, the notation \underline{\lim} (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 infimum article.  (Note: I am not arguing this notation is more common than using \lim \inf, I am merely saying that it is commonly used enough to be worth mentioning). Adammanifold (talk) 19:20, 12 February 2010 (UTC)
thread_title:  Notation for lim inf 

message: They are put on wrong places. Should be swapped.  —Preceding unsigned comment added by 146.50.196.179 (talk) 18:38, 6 October 2010 (UTC)
replies: text: I noticed this too and fixed it. Hashproduct (talk) 18:25, 1 December 2010 (UTC)
thread_title:  Superadditivity and subadditivity 

message: Any relationship between liminf A_n B_n   and liminf A_n    and liminf B_n? Jackzhp (talk) 21:59, 13 January 2011 (UTC)
thread_title: Set

message: 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. 
Coolkid70 (talk) 17:38, 21 January 2011 (UTC)
thread_title:  Properties Section Unreadable 

message: 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 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.  — Preceding unsigned comment added by 66.15.180.216 (talk) 17:43, 5 November 2011 (UTC)
replies: text: 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 xn. Toolnut (talk) 07:16, 15 December 2011 (UTC)
thread_title:  Bad graphic 

message: 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 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.
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. 208.107.152.253 (talk) 23:35, 11 November 2011 (UTC)
thread_title:  Alternate definition 

message: I do not understand the idea with the sentence
::More generally, these definitions make sense in any partially ordered set, provided the suprema and infima exist, such as in a complete lattice.
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, or the sense of the sentence should be the same as for the shorter
::More generally, these definitions make sense in any complete lattice.
On the other hand, there are 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 countable (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. JoergenB (talk) 03:25, 23 May 2013 (UTC)
thread_title:  Unnecessary complicated? 

message: It appears that this is very recently proved in the affirmative. https://www.simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/  — Preceding unsigned comment added by 67.242.141.205 (talk) 03:48, 24 May 2013 (UTC)
replies: text: In order to make your foonote visible, I add <references>. JoergenB (talk) 15:04, 25 May 2013 (UTC)
thread_title:  Update on the question boundedness of prime number gaps 

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thread_title:  External links modified 

message: 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. 
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, 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. 49.198.125.140 (talk) 07:00, 2 August 2019 (UTC)
replies: text: "Limit" is correct; the "inferior" and "superior" are postpositive adjectives.--Jasper Deng (talk) 07:43, 2 August 2019 (UTC)
replies: text: @Jasper Deng: 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. 𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 (𝗍𝗮𝘭𝙠) 10:41, 7 March 2021 (UTC)
replies: text: Look at literally any of the English sources cited here and you’ll see this is he prevailing usage in English. The WP:BURDEN is on you, as the assertor of a contrary claim, not me, to show that the prevailing usage in English is “limes”.—Jasper Deng (talk) 18:59, 7 March 2021 (UTC)
replies: text: @Jasper Deng This was not what I said. The question at hand is whether the abbreviations \limsup and \liminf stand for "limit" or "limes" superior/inferior. 𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 (𝗍𝗮𝘭𝙠) 19:02, 7 March 2021 (UTC)
replies: text: 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 Wikipedia doesn’t care.—Jasper Deng (talk) 19:14, 7 March 2021 (UTC)
thread_title:  Limit or Limes? 

message: :The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this section. 
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:Limit superior and limit inferior →  – For consistency with Infimum and supremum. This is both the alphabetical order and the order from smaller to greater. 𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 (𝗍𝗮𝘭𝙠) 20:56, 27 February 2021 (UTC)
thread_title:  Requested move 27 February 2021