I tried this problem quite a bit but went nowhere. I wish to solve this using set-theoretic algebra.
Problem statement
Let $A_n$, $n \geq 1$ be a sequence of events such that $ \mathbb P(A_n) \to 0$ as $n \to \infty$, and
$$\sum_{n=1}^\infty\mathbb P (A_{n+1}\setminus A_{n}) <\infty$$
Show that almost surely, only finitely many of the $A_n$s will occur.
While irrelevant, it might be worth mentioning this problem is from MIT OCW (Exercise 3).
My attempt
I did not get very far. While it is obvious that we have to use Borel-Cantelli lemma on the hypothesis, it did not lead me anywhere. Specifically, since we know that
$$\sum_{n=1}^\infty\mathbb P (A_{n+1}\setminus A_{n}) <\infty$$
we can conclude that
$$\mathbb P \left ( \bigcap_{n = 1}^{\infty}\bigcup_{k = n}^{\infty} A_{n+1}\setminus A_{n} \right ) = 0$$
Using the above and the hypothesis that $ \mathbb P(A_n) \to 0$ as $n \to \infty$, we need to somehow show that
$$\mathbb P \left ( \bigcap_{n = 1}^{\infty}\bigcup_{k = n}^{\infty} A_{k} \right ) = 0$$
I tried a lot of set algebra but it did not lead me anywhere.
I did see a question which is similar to this one but the hypothesis is slightly different. It mentions
$$\sum_{n=1}^\infty\mathbb P (A_{n}\setminus A_{n+1}) <\infty$$
instead of
$$\sum_{n=1}^\infty\mathbb P (A_{n+1}\setminus A_{n}) <\infty$$
as given in this problem. More importantly, I wish to solve this problem using set-theoretic algebra rather than arguments on lim sup etc.
I would be grateful if you can provide some insight or a hint in order for me to prove this using set-theoretic methods (ie using infinite unions, intersections, De-Morgan's laws etc).
Note that $$ \begin{align} P(\cup_{k=n}^\infty A_k) &=[P(A_n)+ P(A_{n+1}A_n^c)+P(A_{n+2}A_{n+1}^cA_n^c)+\dotsb]\\ &\leq P(A_n)+\sum_{k=n}^\infty P(A_{k+1}\setminus A_{k})\to 0 \end{align} $$ as $n\to \infty$. The result follows from measure continuity.