Hitting time stochastic process

Question

$(X_t)_{t\geq 0}$ is a stochastic process with right continuous sample paths. $U\subset\mathbb{R}$ open set, $\tau_{U}:=\inf\{t>0:X_t\in U\}$ ($\inf \emptyset :=\infty $). Show thar for all $t\geq0$ $\{\tau_U<t\}\in\mathcal F_t^X$, and $\{\tau_U\leq t\}\in \mathcal F_{t+}^X$. I've tried many paths, but I haven't succeeded :(. How can I solve it?