Prove that $P(\bigcup_{i=1}^{\infty} A_i) = 1$ $A_i$ $(i=1,2,...)$ are independent events
$\sum_{i=1}^{\infty}P(A_i) = \infty.$ 
Prove that: 
$P(\bigcup_{i=1}^{\infty} A_i) = 1 $
Can someone please help me out with this question?

By virtue of the non-trivial part of Borel-Cantelli lemma, one is allowed to conclude that the event $\displaystyle \mathrm{A} = \{A_n, \mathrm{i.o.}\} = \bigcap_{k = 1}^\infty \bigcup_{n = k}^\infty A_n$ has total mass equal to one. Notice that $\mathrm{A}$ lies within the union of the $A_n$ and the exercise follows.
