This is a soft question in the sense that I don't have a particular problem in mind, but rather, I have a general confusion:
I understand that (one) of the advantages of using $\liminf$ and $\limsup$ of a sequence or a function is that they always exist (as opposed to the $\lim$ which may fail to exist). However, many times I find myself asking the question: why take a $\liminf$ and not a $\limsup$?.
What is the difference between them? Or why choose one and not the other for some computation? or for some limiting process etc.
I know this is a very vague question, and therefore, you are welcome to interpret it as you like. I just want to start to clarify this confusion that I have had for a long time.
It used to describe sequences of events which say that under certain conditions the probability that infinitely many events happen is either zero or one. They are used in the Borel-Cantelli lemmas, which is important since they are closely related to almost sure convergence, thus help us understand it better. They also provide a powerful tool when proving almost sure convergence.
Let $\{A_n : n \in \mathbb{N}\}$ be a sequence of events from a sample space $\Omega$.
If $\omega \in \bigcap_{n=1}^\infty \bigcup_{m=n}^\infty A_m$, then for every $n\ge 1$ there exists an $m\ge n$ such that $\omega \in A_m$. Thus for $\omega \in A_m$ for infinitely many values of $m$, or in other words, $A_m$ happens infinitely often.
On the other hand, if $\omega \in \bigcup_{n=1}^\infty \bigcap_{m=n}^\infty A_m$, there exists an $n\ge 1$ such that $\omega \in A_m$. That means that starting from $n$, all $A_m$, $m\ge n$, happen, which also mean that only finitely many $A^c_n$ happen. We say that $A_m$ happen eventually.
Thus, $$ \{\lim \sup A_n\}^c = \{\lim \inf A_n^c\} $$