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?

TL;DR BCL2 tells us $A_n$ occurs infinitely often. So obviously, at least one $A_n$ occurs.

Pf:
$$\sum_{i=1}^{\infty}P(A_i) = \infty$$
and
$A_n$ indp
By BCL2, we have $P(\limsup A_n) = 1$
Note that $$\limsup A_n \subseteq \bigcup_n A_n \tag{*}$$
QED

Intuition:
$(*)$ should be not so difficult to prove and should be easily understood intuitively:
In the sequence
$$A_1, A_2, A_3, A_4, ...$$
we deduce by BCL2 that for any day $m$, (let's say $A_m$ corresponds to day $m$), there is some future day $n \ge m$ that will occur eg on day 1, there is some day $n \ge 1$ s.t. $A_n$ will occur. On day $m=100$, there is some $n \ge 100$ s.t. $A_n$ will occur. We say that $A_n$ occurs infinitely often.
So if $A_n$ occurs infinitely often, then obviously at least one of those $A_n$'s will occur i.e. $$P(\bigcup_{n=1}^{\infty} A_n) = 1$$
