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I am recently studying limit superior & limit inferior of sequence of subsets. Then I came across this phrase. What does this actually mean? I have understood "for infinitely many $n$" but am not understanding the above phrase. Can anyone help me explain what the phrase wants to convey?

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    A sequence of sets (or subsets of a given set) ${ X_n }$ is "indexed" like a "usual" sequence (see your previous post ); thus, saying that some property $P$ holds "for all but finitely many" sets of the sequence means that there is $N_0$ such that the property $P$ holds for all $X_n$ such that $n > N_0$. – Mauro ALLEGRANZA Jun 03 '15 at 09:31
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  • In case of wondering what it means in English, I assume a more understandable version would be "all except finitely many", instead of the usual "all but" meaning "almost". – IS4 Nov 09 '20 at 01:02

2 Answers2

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Let's talk about sets of natural numbers as a motivating example. It's pretty clear what it means to have a collection of natural numbers that is "infinite." Beyond that, there is a stronger requirement on such a collection, that it contains "all but finitely many" of them (sometimes called "cofinitely many"). By definition, a set is cofinite if it's complement is finite.

Consider the even numbers. There are infinitely many of them. However, there are also infinitely many odd numbers, so there are not cofinitely many even numbers. On the other hand, look at the natural numbers $> 1000$. There are infinitely many of them. There are also cofinitely many of them, since there are only finitely many natural numbers $\leq 1000$.

Hopefully that illustrates how "cofinitely many" (i.e., all but finitely many) is a stronger requirement than "infinitely many."

Regarding limit superior and limit inferior of a collection of sets, the limit superior contains those points which are in infinitely many sets of the collection, and the limit inferior contains those points which are in cofinitely many of the sets of the collection. Since cofinite is a stronger requirement than infinite, that means that it's harder for a point to be in the limit inferior, so the limit inferior is a subset of the limit superior.

  • That means given a condition, infinite elements satisfy that while there are also finite number of elements that contradict the condition, right, sir?+1 for the 1000 example. –  Jun 03 '15 at 09:21
  • @user36790 Yes, so long as you mean to say that there are also only a finite number of elements that contradict. –  Jun 03 '15 at 09:24
  • Can you tell me just why is the definition of limit superior & limit inferior like this? Yes, I can easily understand the corresponding definitions for sequence of real numbers but never could conceive why it is so in the case for sets. –  Jun 03 '15 at 09:25
  • Then how can I let you make me help: I had posted a question earlier that asked about the same. Can you please help me in that? –  Jun 03 '15 at 09:32
  • @user36790 One way to see that they're related is this. Assume you have your collection of sets, and fix some point $x$. Define a sequence of 1s and 0s, with a 1 whenever $x$ is in the set, and a 0 whenever it's not. The $\limsup$ of this sequence is 1 just if the point is in the $\limsup$ of the sets, and similarly for $\liminf$. They're both really the same underlying concept. –  Jun 03 '15 at 09:33
  • You are talking of indicator function, right? I got it in the stats class. –  Jun 03 '15 at 09:37
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    @user36790 Yes. The indicator function is also sometimes called the "characteristic function." –  Jun 03 '15 at 09:42
  • Thanks for patiently answering my queries; I have to concise all the gathered info now & think a lot. Thanks again:) –  Jun 03 '15 at 09:45
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There may be three or ten or twenty thousand exceptions, but eventually they run out.

If there were infinitely many exceptions, there would always be an exception greater than any arbitrary integer $N$. But if there are only finitely many exceptions, you can find an integer $N$ beyond which there are no more.

Mark Bennet
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  • Sir, by "exceptions", what do you want to mean? I am a bit new to this topic , so ...... Also, what does this meaning of the phrase correspond to the definition of limit inferior? –  Jun 03 '15 at 06:52
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    @user36790 The definition of lim inf = L goes roughly that for any $\epsilon\gt 0$, everything you are considering is eventually greater than $L-\epsilon$ - so either greater than $L$ or only a small distance below it. The exceptions are the initial values which may be less than $L-\epsilon$ - the finite number of values which are "a long way" below $L$. – Mark Bennet Jun 03 '15 at 07:27
  • Sir, for a clearer picture, would you please define in the same way the limit superior highlighting "for infinitely many $n$" so that I may make contrast between the two? I'll be grateful. –  Jun 03 '15 at 07:47
  • @user36790 Take $L$ as the upper limit and use $L+\epsilon$ - the exceptions are the early values which are too large – Mark Bennet Jun 03 '15 at 10:56