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:
$A_n=\left\{\frac0n, \frac1n, \dots , \frac{n^2}n\right\}$
and I am trying to find the limsup and liminf.
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 $\liminf A_n\subset\limsup A_n$.
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 $n\to\infty$, all of the positive rational numbers appear. I am having trouble seeing the limits. For example, no matter how large I choose $N$, there is some $n\ge N$ 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.
One simple question: does an event have to "not" show up sometimes to be part of the liminf, or is it just that it is allowed to be missing finitely often? Assuming for a moment that the former is true, it appears to me that {0} is definitely in both liminf and limsup: that is, no matter how large I select N, {0} is in some (in this case, all) A_n with n>N.
Moving on from there, it is clear to me that the integers begin to appear over and over again, and as n-->infinity, the rationals begin to "fill out" as well. Where I seem to be getting stuck is that the integers only show up equal to "n" and it is not clear to me how to handle the fact that the sequence is unbounded. Intuitively, all of the (positive) integers eventually show up (always), but they are far from the only values that do. {1/2} shows up always, for example. I'm not sure if {Q_+} eventually shows up, and this may be due to a lack of formal construction of Q in my past. What I recall is that Q is essentially all real numbers that can be expressed as m/n with m and n members of Z.
Thank you.