Proving an implication in probabilty

Question

Being $(\Omega,F,P)$ a probability space and {$A_i$} an independent set sequence with $A_i \in F$, $i=1,2,... $

How can I prove that if $\sum_{i=1}^\infty P(A_i) =\infty$, then

$$\lim_{n\to \infty} P\bigl(\cap_{i=1}^n \cup_{k=1}^\infty A_k\bigl)=1$$

Answer 1

This is the second Borel Cantelli Lemma. You can find the proof in most introductory probability textbooks or on Wikipedia.