I am trying to understand the following implication in the proof of Borel-Cantelli. To set it up for you, we are given:
Let $A_i$ be a seq. of events and $\lim_{n \rightarrow \infty }\sup A_n = \bigcap_{n\geq1}\left(\bigcup_{m\geq n} A_n \right)$. The statement is, if $\sum_{n\geq 1}P(A_n) < \infty $ then $P(\limsup A_n)=0$.
My question is, why does $\sum_{n=1}^{\infty}{1_{A_n}} < \infty \;P-a.s.$ imply that $P(\limsup (A_n)) = 0$?
And if I have explicitly $A_n$ given as the open interval $(0, \frac{1}{n})$ with sample space $\Omega=(0,1)$, sigma algebra $=\sigma((0,1))$ and the Lebesguemeasure on $\Omega$, why it then $lim sup(A_n)=0$?
Thank you.
Hint for your second question, expand: $$\lim_{n \rightarrow \infty }\sup A_n = \bigcap_{n\geq1}\left(\bigcup_{m\geq n} A_n \right)=(A_1\bigcup A_2\bigcup A_3\bigcup ...\bigcup A_n...)\bigcap\bigcap(A_2\bigcup A_3\bigcup ...\bigcup A_n...)\bigcap(A_3\bigcup A_4\bigcup A_5\bigcup ...\bigcup A_n...)\bigcap...\bigcap(A_n\bigcup A_{n+1}\bigcup A_{n+2}\bigcup ...)\bigcap...$$
Now, if $A_n=(0,\frac{1}{n})$ try to see which element(s) will be in ALL intersecting gaps $(\cdot)$