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?