Assuming the usual conditions hold and that X ist càdlàg, how can we proof, that for a closed subset $A\subset \mathbb{R}$ the random variable
$\tau := \inf\{ t\geq 0: X_t\in A \mbox{ or} X_{t-}\in A\}$
is a stopping time?
Edit, a little bit more explanation:
So because $A$ is closed, we get $$X_t(\omega )\in A \Leftrightarrow \text{dist}(X_t(\omega),A)=0.$$ And for the left limes $X_{t-}$ we get $$X_{t-}(\omega )\in A \Leftrightarrow \lim\limits_{ t_n \uparrow t, t_n\not=t } \text{dist}(X_{t_n}(\omega), A)=0$$ because $X$ is càdlàg. So it is correct to write (and because the usual conditions hold we only need to look at $\{ \tau < t\}$) $$\{ \tau < t\} = \{ \inf\limits_{ u\in [0,t) } \text{dist}(X_u, A) = 0\}.$$ And we can rewrite this: $$ \left\{ \inf\limits_{ u\in [0,t) } \text{dist}(X_u, A) = 0\right\} = \bigcap_{n\in \mathbb{N}} \left\{ \inf\limits_{ u\in [0,t) } \text{dist}(X_u, A) < 1/n \right\} \\=\bigcap_{n\in \mathbb{N}} \left\{ \bigcup_{ u \in \mathbb{Q} \cap [0,t) } \left\{ \text{dist}(X_u, A) < 1/n \right\} \right\}.$$ And because of $\left\{ \text{dist}(X_u, A) < 1/n \right\} \in \mathcal{F}_u \subset\mathcal{F}_t$ we get $$\bigcap_{n\in \mathbb{N}} \left\{ \bigcup_{ u \in \mathbb{Q} \cap [0,t) } \left\{ \text{dist}(X_u, A) < 1/n \right\} \right\} \in \mathcal{F}_t,$$ So $\left\{ \tau < t \right\}\in \mathcal{F}_t$. So $\tau$ is a stopping time.