Prove probability of event i.o. equal to 0 with given conditions

Question

Suppose events $\left\{A_n\right\}_{n=1}^\infty$ are from a common probability space. And we have
$$\lim_{n \rightarrow \infty}P(A_n)=0,\qquad\sum_{n=1}^\infty P(A_nA_{n+1}^c)< \infty$$
Prove: $P(A_n \text{ i.o.})=0$.

I think it can be proved using Borel-Cantelli lemma. But not sure how to show $\sum_{n=1}^\infty P(A_n)< \infty$

Answer 1

Hint: show that the event

$$\left(\bigcup_{k \ge n} A_k\right) \triangle \left(\bigcup_{k \ge n} (A_k \cap A_{k+1}^c)\right)$$ has zero probability. Here, the symmetric difference is defined by $B \triangle C := (B\cup C) - (B\cap C)$.

This will allow you to transfer the result of Borel-Cantelli on $\{A_n \cap A_{n+1}^c\}_{n=1}^\infty$ to $\{A_n\}_{n=1}^\infty$.

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Edit:

The above symmetric difference is contained in $\bigcup_{k \ge n} \bigcap_{j \ge k} A_j$. [Check this.]

Then, \begin{align} P\left\{\left(\bigcup_{k \ge n} A_k\right) \triangle \left(\bigcup_{k \ge n} (A_k \cap A_{k+1}^c)\right)\right\} &\le P\left(\bigcup_{k \ge n} \bigcap_{j \ge k} A_j\right)\\ &= \lim_{k \to \infty} P\bigcap_{j \ge k} A_j\\ &\le \lim_{k \to \infty} P(A_k)\\ &\to 0. \end{align}