Consider a submartingale $X,$ then for almost every $\omega \in \Omega,$ for every $v \in \mathbb{R},\lim_{u\in \mathbb{{Q},u \uparrow v}}X_u(\omega)$ exist in $\mathbb{R}.$
I was wondering if there are proofs in the literature for this theorem without applying the upcrossing inequality. I know various proofs for the discrete case, but I didn't find a particular one for the above.