$P(\bigcup_{n=1}^\infty A_n)=1$ iff $P(A_n i.o.)=1$
I have always struggled with the "infinitely often" and "all but finitely often" concepts for sequences. My first time through Resnick, I don't think I was equipped with the tools necessary for this exercise using only what I had learned in the text previously, and so I wanted to come back to a few problems.
I still am not sure how to proceed at all here.
I think I understand the question to basically state "The probability that a union of a sequence of events is 1 if and only if the probability of that sequence occurring infinitely often is also 1.
As per the comments below, an additional assumption is that each of the events has $P(A_i)<1$ and each event is independent.
So, to start out, I can define:
$\{A_n i.o.\}=\limsup_{n\to\infty}A_n=\bigcap_{n=1}^\infty\bigcup_{k=n}A_k$
but from here I am not sure where to go (or if I've even begun correctly).
Feel free to be as explicit as possible as these ideas ($\limsup$, etc.) have always seemed to elude me. Thank you for the help.
Best Answer
Hint Show that (1) $\iff$ (2) $\iff$ (3) $\iff$ (4) $\iff$ (5) $\iff$ (6), where:
(1) $\mathbb P\left(\bigcup\limits_{n\geqslant1}A_n\right)=1$. (2) $\mathbb P\left(\bigcap\limits_{n\geqslant1}(\Omega\setminus A_n)\right)=0$. (3) $\prod\limits_{n\geqslant1}(1-\mathbb P(A_n))=0$. (4) $\sum\limits_{n\geqslant1}\mathbb P(A_n)=+\infty$. (5) $\mathbb P\left(\limsup A_n\right)=1$. (6) $\mathbb P\left(A_n\ \text{i.o.}\right)=1$.
If a step is unclear, please say so.