I want to show that some sequence of events $A_n$ happens almost always, that is $\mathbb{P}(\{A_n\} \text{ a.a.}) = 1$, which is equivalent to showing $\mathbb{P}(\{A_n^C\} \text{ i.o.}) = 0$. I could use Borell-Cantelli's second lemma, but there is one problem:
The following series does not necessarily converge: $\sum_{n=1}^\infty \mathbb{P}(A_n^C) $. What we do now however, is that $\lim_{n\rightarrow\infty} \mathbb{P}(A_n^C) = 0$.
Thus my intuition tells me that perhaps convergence in probability could play a role here, but then the question arises: To what does it converge to in probability then?