Let $A_n$ be a sequence of events in probability space.
The event $A(i.o.)$=$\{A_n$ happens infinitely often $\}$ is formally stated as $\lim \sup A_n$ (in other words $\cap_{k=1}^\infty \cup_{n=k}^\infty A_n$).
This is because if an element belongs to $\lim \sup A_n$, it can be shown that it belongs to infinitely many $A_n$ (because it belongs to all sets $\cup_{n=k}^\infty A_n$).
In the proof of Borel-Cantelli lemma on page 2, it says:
$P(A(i.o.)) = \lim_{k \to \infty}P(\cup_{n=k}^\infty A_n)$.
According to the definition above, it's exactly the same as
$$P(\cap_{k=1}^\infty \cup_{n=k}^\infty A_n)= \lim_{k \to \infty}P(\cup_{n=k}^\infty A_n)$$
Could anyone prove this equality?