# Proving Borel Cantelli Like Result

Question

Suppose that $$(A_n)_{n\geq 1}$$ is a sequence of events such that $$P(A_n)\to0\quad \text{and}\quad \sum_1^\infty P(A_n\setminus A_{n+1})<\infty.$$ Prove that $$P(A_n \, \text{i.o})=0$$ where $$(A_n \, \text{i.o})=\cap_{n=1}^\infty\cup_{k=n}^\infty A_k$$.

My attempt

I don't think I can apply Borel cantelli directly since I believe that the condition in the question is a sharper result. I have been able to prove the result in a similar scenario when $$P(A_n)\to0$$ and $$\sum_1^\infty P(A_{n+1}\setminus A_n)<\infty$$. Indeed in this case \begin{align} P(\cup_{k=n}^\infty A_k) &=[P(A_n)+ P(A_{n+1}A_n^c)+P(A_{n+2}A_{n+1}^cA_n^c)+\dotsb]\\ &\leq P(A_n)+\sum_{k=n}^\infty P(A_{k+1}\setminus A_{k})\to 0 \end{align} as $$n\to \infty$$.

The condition $$\sum_1^\infty P(A_n\setminus A_{n+1})<\infty$$ reminds me of nested decreasing intersections but I have been unable to rewrite the event $$(A_n \, \text{i.o})$$ in order to use it. Any help is appreciated

Verify that $$\{A_n i.o. \} \subset \{A_n \setminus A_{n+1} i.o \} \cup B$$ where $$B$$ is $$\lim \inf A_n$$ (i.e. points that belong to $$A_n$$ for all $$n$$ sufficiently large. Note that $$P(B)=0$$ because $$P(A_n) \to 0$$. Let me know if you need more details.