If $P(A_n) \rightarrow 0$ and $\sum_{n=1}^{\infty}{P(A_n^c\cap A_{n+1}})<\infty$ then $P(A_n \text{ i.o.})=0$.
How to prove this? Thanks.
If $P(A_n) \rightarrow 0$ and $\sum_{n=1}^{\infty}{P(A_n^c\cap A_{n+1}})<\infty$ then $P(A_n \text{ i.o.})=0$.
How to prove this? Thanks.
Hint: $\lim \sup A_n \subseteq A_N \cup \bigcup_{n=N}^\infty (A^c_n \cap A_{n+1})$. Estimate the probability of this.