Let $(\mathcal{F}_n)_{n\in\mathbb{N}}$ be a sequence of $\sigma$-algebras and $A_n\in\mathcal{F}_n$ $\forall n$.
Then, if $\sum\limits_{k=1}^n \mathbb{E}[\mathbb{1}_{A_k}|\mathcal{F}_{k-1}]\stackrel{n\rightarrow\infty}{\longrightarrow}\infty$,
$$ \frac{\sum\limits_{k=1}^n \mathbb{1}_{A_k}}{\sum\limits_{k=1}^n \mathbb{E}[\mathbb{1}_{A_k}|\mathcal{F}_{k-1}]}\stackrel{n\rightarrow\infty}{\longrightarrow}1 $$
Showing this is an exercise in the context of martingale theory, I'm assuming I should be able to identify a martingale somewhere. But I'm struggling to find it, the arbitrariness of the $\sigma$-algebras and the sets stump me.
Any hints towards the solution and, as an extra, general intuition* about it would be much appreciated.
*i.e., if you see it (quickly), what's the more general pattern of the martingale here which may pop up someplace else?