# Borel Cantelli lemma question part 2

Earlier I had posted question about Borel cantelli Lemma. Thanks to Nate and Shai for your kind response. Please see my profile for the earlier questions. As you can see I do not understand the lemma well. Here are some confusions I have. I hope someone will be kind enough to help me understand the following:

Is it true that P ([An i.o.]∩ A) = P [( A ∩ An) i.o.]. If so, how do we prove this? or is it trivial? And assuming that we know that P[An i.o.] =1 , can we establish that the sum {P(A ∩ An)} diverges (meaning = infinity) ?

Thanks

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The events "$A_n$ for infinitely many $n$; and $A$" and "$A_n \land A$ for infinitely many $n$" are one and the same since $\land$ distributes over $\lor$: $$A \land \bigwedge_{n \geq 0} \bigvee_{m \geq n} A_m \Leftrightarrow \bigwedge_{n \geq 0} A \land \bigvee_{m \geq n} A_m \Leftrightarrow \bigwedge_{n \geq 0} \bigvee_{m \geq n} A \land A_m.$$
As for your second question, if $A$ never happens then $P(A \land A_n) = 0$.