I am having trouble with a step in the proof of Theorem 2.9.6 of Karatzas and Shreve's Brownian Motion and Stochastic Calculus. I need to prove that, given a Brownian motion $W$, $$\beta_t:=\inf\{s>t; W_s=0\}$$
is an optional time for every $t\ge 0$.
Here is my attempt: fix $t\ge0$, and let $T \in \mathbb{R}$. If $T \le t$, $\{\beta_t <T\}=\emptyset\in\mathcal{F}_T$. Otherwise, $$\{\beta_t<T\}=\bigcup_{s<t'<t}\{W_{t'}=0\}\in \sigma\left( \bigcup_{s<t'<t}\mathcal{F}_{t'}\right)\subseteq \mathcal{F}_t.$$
Is this correct? If not, there is some hint I could follow? Thank you!
I believe I solved the problem. Given $T>t$, we have that
$$\{\beta_t \ge T\}=\bigcap_n A_n,$$
where $A_n=\{W_s\neq 0; s\in[t+1/n,T-1/n]\}$. But since $W$ is continuous and $[t+1/n,T-1/n]$ is closed, we have $$A_n = \bigcup_{k\ge 1} \bigcap_{q\in[t+1/n,T-1/n]\cap\mathbb{Q}}\{|W_q|\ge1/k\} \in \mathcal{F}_T.$$
Therefore $\{\beta_t<T\}=\{\beta_t \ge T\}^c\in \mathcal{F}_T$.