Lemma:If $X$ be a right-continuous positive local martingale then , $X$ is a generalized super martingale Proof: $\forall s<t$
$$E[X_t\mid F_s]=E[\lim_{n\to\infty} X_{t \wedge\tau_n}\mid F_s] \leq \liminf_{n \to \infty }E[X_{t \wedge\tau_n}\mid F_s]= \liminf_{n\to \infty} X_{s\wedge\tau_n}=X_s$$
I fail to see where the assumption of right continuity being used. It often happens that in proves of continuous time processes , I fail to see where the assumptions are being used unless explicitly mentioned in the text Thank you EDIT: Its a positive martingale?