similar to second borell cantelli lemma

Question

Let $A_n$ be a sequence of events in a probability space $(X,F,P)$ such that $P(A_n)<1$ for all $n$ , $ P(\bigcup A_n)=1$ , and $A_n$ are independent. Prove that $P(A_n i.o)=1$

Answer 1

First and foremost, let it be clear that all we have to do is show that:

Then, by the 2nd Borel-Cantelli Lemma and the independence of the sequence of events, we have the result. Let's begin:

Because of the independence:

The two previous equations give us:

At this point, you have to be familiar with infinite product of sequences. If you are not, then let me know so i can inform you, or google some information about it. There is a statement, which i found in a complex analysis textbook, that makes this problem a lot easier:

Corollary: Let be a sequence of real numbers, such that . Then:

which (since all the terms are non-negative) is the same as this:

Now, we apply the result above for:

We are able to do this because we know that:

So, from the corollary, we have:

and finally, because of the above, from the 2nd Borel-Cantelli Lemma, we get: