Let $X$ be a stochastic process with every sample path RCLL(right continuous on $[0,\infty)$ and with left-hand finite limits on $(0,\infty)$). Let $A$ be an event that $X$ is continuous on $[0,t_0)$. Show $A\in\mathcal{F}_{t_0}^X$.
For $(t_n)_n$ a dense sequence contained in $(0,t_0)$, if we can show
$$A=\{\omega: \lim_{s\rightarrow t_n-}X_s(\omega)=X_{t_n}(\omega),\ \ \forall n\geq 1,\ \ t_n<t\}$$
then $A\in\mathcal{F}_{t_0}^X$ since every $\{\omega: \lim_{s\rightarrow t_n-}X_s(\omega)=X_{t_n}(\omega)\}\in\mathcal{F}_{t_0}^X$. Could someone show how to prove
$$A=\{\omega: \lim_{s\rightarrow t_n-}X_s(\omega)=X_{t_n}(\omega),\ \ \forall n\geq 1,\ \ t_n<t\}$$