# probability theory – If a process is right-continuous why can I approximate with sequences \$(tau_{n}land t)\$ , such that \$tau_{n}uparrow infty\$

Consider a process $$(X_{t})_{tgeq 0}$$ that is a local martingale with following reducing stopping times $$(tau_{n})_{nin mathbb N}$$, I want to know whether the right-continuity of the process is indeed necessary for the statement $$limlimits_{n to infty}X_{tland tau_{n}}=X_{t}$$ for any $$t$$.Or could the assumption of right-continuity be replaced by left-continuity? I am just unsure why right-continuity suffices in this case because surely the sequence $$tland tau_{n}leq t$$ so we are approaching $$t$$ indeed from the left. So why is right-continuity used?