I'm working through a proof outline of the generalized Borel-Cantelli lemmas. Following the outline, I've shown that for $A_k\in\mathcal F_k$ and $$X_n=\sum_{k=1}^n 1_{A_k}, Y_n=\sum_{k=1}^n P(A_k\vert \mathcal F_{k-1})$$ we have that $M_n=X_n-Y_n$ is a martingale with $E(M_n^2)<\infty$ and $\langle M_n\rangle\le Y_n$ for all $n$.
Now the missing step is to show that $X_\infty<\infty\iff Y_\infty<\infty$ a.s., and the proof outline says to consider first $\{\langle M_\infty\rangle < \infty\}$ and then $\{\langle M_\infty\rangle = \infty\}$
I already know that the limit of $M_n$ exists a.s. on the set $\{\langle M_\infty\rangle<\infty\}$, so because $X_n, Y_n$ are non-negative, on that set the equivalence clearly holds.
However on the set $\{\langle M_\infty\rangle = \infty\}$ I also know that the limit doesn't exist a.s., but it is not self-evident why this implies the equivalence (at this point it might just go to $\infty$ or oscillate between finite values).
Usually, this entire proof is done using the fact that a finite increment martingale either converges or oscillates between $-\infty$ and $\infty$. I understand the proof of this property. However, if I show this property, then I don't actually need to consider the sets $\{\langle M_\infty\rangle < \infty\}$ and $\{\langle M_\infty\rangle = \infty\}$! I also don't need the properties I've shown before, just that $M_n$ has bounded increments.
So I'm wondering if I'm missing a much easier proof of this statement given what I have already shown. Does someone see a connection I'm missing?