I'm trying to prove the next:
Let $X$ be an adapted process and right-continuous. Let $\tau$ be a stopping time. Then there is a sequence of discrete stopping times $(\tau_{n})_{n\in\mathbb{N}}$ which converges to $\tau$ and satisfies that $$\displaystyle\lim_{n\rightarrow\infty}X_{\tau_{n}}=X_{\tau}.$$
So, for any stopping time $\tau,$ there is a decreasing sequence of discrete stopping time $(\tau_{n})_{n\in\mathbb{N}}$ such that $\tau_{n}\rightarrow\tau$ when $n\rightarrow\infty.$ Such sequence can be constructed as follow:
$$\tau_{n}(\omega) = \begin{cases} \tau(\omega) & \text{if}\space\tau(\omega)=\infty; \\ k2^{-n} & \text{if}\space(k-1)2^{-n}\leq\tau(\omega)<k2^{-n}.\end{cases}$$
The step in which I'm stuck is on proving $X_{\tau_{n}}\rightarrow X_{\tau}.$
Any kind of help is thanked in advanced.