I want to prove that that supremum of a integrable nonnegative supermartingale $M^n$ is finite.
By the martingale convergence theorem, we know that $\lim_n M^n = M$ almost surely with $E[M]<\infty$. I think we can argue that finite limit implies finite supremum but M being a random variables confuses me. Can we make this argument even though $M$ is a random variable, not a fixed number?