The second Borel-Cantelli lemma refers to a sequence of independent events $A_n$ such that $\sum_{n=1}^\infty \Pr(A_n)=\infty,$ and says that in this case infinitely many $A_n$ events occur almost surely.
$$\bigcap_{n=1}^\infty \bigcup_{i=n}^\infty A_i=\{A_n \text{ i.o.}\}$$
The indexing in the union considers the union of only events beyond a certain event $n$ in the sequence. So it would comprise the occurrence of any events after $n.$
But what is the intersection? Does it simply mean that all these composite events after every single $n$ do occur?
Best Answer
Just read the intersection as a "for all", and the union as a "there exists". The event $\bigcap_{n=1}^\infty \bigcup_{i=n}^\infty A_i$ then means : "for all $n \geq 1$, there exists $i \geq n$ such that the event $A_i$ occurs". In other words, this means indeed that infinitely many events $A_i$ occur.