# Liminf and limsup of sequence of sets

1. May 19, 2012

### jbalthrop

1. The problem statement, all variables and given/known data

I am attempting to learn some measure theory and am starting with liminf and limsup of sequences of sets.

I found an example that is as follows:

A_n={0/n, 1/n, ... , n^2/n} and I am trying to find the limsup and liminf.

2. Relevant equations

liminf \subset limsup

3. The attempt at a solution

My understanding is that both deal with the tail sequences, and that limsup involves values that appear "infinitely often" and liminf covers values that appear "all but finitely often". Also I understand that lim infAn⊂lim supAn.

For the above example, if I enumerate the first few sets, it is clearly evident that {0} appears i.o. It also seems (to me) that as n--> infinity, all of the positive rational numbers appear. Now, since the sequence is monotone non-decreasing, An⊂An+1, I am having trouble seeing the limits. For example, no matter how large I choose N, there is some n≥N in which all of the rationals appear, right?

Obviously I am confused (this is all self-taught), so any explanation would be greatly appreciated. I seem to be able to make sense of liminf and limsup when the sequence is of a form similar to [0, n/n+1) and other examples, but I'm struggling with this example.

Thank you.