What does "for all but finitely many $n$" in the definition of limit inferior mean? 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?
 A: 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.
A: 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.
