I have this problem.
Let $X_t$ be a continuous, adapted stochastic process. We define this sequence of stopping times $$ s_n(\omega) = T \land inf\{t\in[0,T]| |X_t(\omega)|\ge n\} $$ In my book is written that $$ \lim_{n\to\infty} s_n(\omega) = T a.s $$
I can't understand that almost surely. I can't figure out a trajectory in which that sequence does not converge increasingly.
Take into account that $inf\{\emptyset \}= \infty$ so that, eventually, $s_n(\omega)=T$.
Thanks !
If $\omega$ is a sample point such that $[0,T]\ni t\mapsto X_t(\omega)\in\Bbb R$ is continuous, then $\lim_ns_n(\omega) =T$. In fact, there exists $n_0(\omega)$ such that $s_n(\omega) =T$ for all $n\ge n_0(\omega)$.