Here are two theorems that I've come across which explain when a martingale converges almost surely:
- If $\{X_n\}$ is a martingale bounded above or bounded below, then $\lim_{n \to \infty}X_n = X$ a.s
- If $\{X_n\}$ is a supermartingale bounded below or a submartingale bounded above, then it converges almost surely.
I've learned that a martingale is both a submartingale and supermartingale. Therefore, I believe theorem 2. could be seen as a corollary to theorem 1.
However, if that is the case, then shouldn't it also be true that if we have submartingale bounded below or a supermartingale bounded above then this sequence also converges almost surely? I'm hesitant to conclude this - because intuitively that does not make a lot of sense.