Show $\sum_{n \geq 1}\mathbb{1}_{A_n}(\omega) = \infty$ iff $\sum_{n \geq 1} \mathbb{E}[\mathbb{1}_{A_n}|\mathcal{F}_{n-1}](\omega) = \infty$

Question

Given a pobability space $(\Omega,\mathcal{A},P)$, a filtration $(\mathcal{F_n})_n$ and a sequence of events $A_n \in \mathcal{F}_n$

I have the martingale $X_n = \sum_{1 \leq j \leq n}(\mathbb{1}_{A_j} - P[A_j|\mathcal{F}_n])$ and I know that $P[B] = 1$, with $B = \{\limsup_nX_n$ exists and is finite$\}$ $\cup$ $\{\limsup_nX_n = \infty, \liminf_nX_n = -\infty\}$

I want to show for $\omega \in B$, that $\sum_{n \geq 1}\mathbb{1}_{A_n}(\omega) = \infty$ iff $\sum_{n \geq 1} \mathbb{E}[\mathbb{1}_{A_n}|\mathcal{F}_{n-1}](\omega) = \infty$

So, I want to exchange a limit and the integral, to achieve:

$$\sum_{n \geq 1} \mathbb{E}[\mathbb{1}_{A_n}|\mathcal{F}_{n-1}](\omega) = \lim_N \sum_{1 \leq n \leq N} \mathbb{E}[\mathbb{1}_{A_n}|\mathcal{F}_{n-1}](\omega) = \lim_N\mathbb{E}[\sum_{1 \leq n \leq N} \mathbb{1}_{A_n}|\mathcal{F}_{n-1}](\omega) = \mathbb{E}[\sum_{n\geq 1}\mathbb{1}_{A_n}|\mathcal{F}_{n-1}]$$

Now my question is, how can I argue, that I can pull the limit into the expectation, if the limit is infinite? Because if the limit were finite, then I could use monotone convergence... but I don't know what to do it it is infinte...

Any help is appreciated :)