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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

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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.

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