# What is the meaning of $\cap_{n=1}^\infty \cup_{i=n}^\infty A_i=\{A_n \text{ infinitely often}\}$ mean?

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?

• The notation is definitely lacking, since the RHS has no reference for what $n$ is. Feb 8 at 17:08
• The left hand side simplifies to $\{x : x \in A_n \text{ for infinitely many } n \in \Bbb{N}\}$, so I'm guessing that's what is trying to be conveyed here. Feb 8 at 17:11
• You could also note that this event is actually the limit superior of the events, which also intuitively matches the "infinitely often". Feb 8 at 19:22

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.