Let $C_1, C_2, \ldots $ be a sequence of events in a probability space
$(\Omega, \mathcal{F}, P)$ such that $\lim_{n\to\infty} P(C_n) = 0$
and $\sum_{n=1}^{\infty} P(C_{n + 1} \setminus C_{n}) < \infty$. Prove
that$$P\left(\bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty} C_{k}\right) =
0.$$
I'm stuck on this problem for a pretty long time. I've seen the following example, which is similar: Probability of intersection of non increasing sequence of events
But it is not enough to help me solve the problem. It seems like there are so many details that I am given, and I am not sure how to use them all.
I would appreciate some sort of help as I have been stuck on this problem for many days now.
Best Answer
Note that by measure continuity from above, $$ P(\cap_{n=1}^ {\infty}\cup_{k=n}^\infty A_k)=\lim_{n\to\infty}P(\cup_{k=n}^\infty A_k). $$ Moreover, countable additivity yields that \begin{align} P(\cup_{k=n}^\infty A_k) &=[P(A_n)+ P(A_{n+1}\cap A_n^c)+P(A_{n+2}\cap A_{n+1}^c\cap A_n^c)+\dotsb]\\ &\leq P(A_n)+\sum_{k=n}^\infty P(A_{k+1}\setminus A_{k})\to 0 \end{align} as $n\to \infty$ where in the last line we use the hypotheses of the problem. The result follows.