Let $A_1, A_2, \ldots$ be events with $A_n\in\mathcal{F}_n$. Show that $$\biggl\{\sum_{n=1}^\infty \mathbf{P}[A_n|\mathcal{F}_{n-1}]=\infty\biggr\} = \limsup_{n\rightarrow\infty} A_n \text{ a. s.}$$
Hint: Consider the martingale $X_n = \sum_{i=1}^n \bigl(1_{A_i} - \mathbf{P}[A_i|\mathcal{F}_{i-1}]\bigr)$.
I don't even know how to start with this exercise. I think one has to use the result that martingales with bounded increments either converge or diverge to $\pm \infty$ almost surely.
Can you give me a hint? Thanks for your time.
