show that infimum is a stopping time

Question

We have a stochastic process $(X_n)_{n\in\mathbb{N}}$ and let $B=[0,1]$. Show that $$\tau=\inf\left\{n \ : \ X_n\in B\right\}$$ is a stopping time.

I know that the minimum of two stopping times is a stopping time but it is the first time with infimum. Firstly, since infimum has not to be attained, I don't even know if we can use, somehow, a formula $\left\{\tau \leq t\right\}$. On the other hand, since $B$ is a finite set, intuitively it seems legit to being 'reduced' to the case with minimum (but only on the intuition level).

Any hints?

Answer 1

What is your definition of stopping time?

For my definition respectivly to the filtration generated by the stochastic process:

For $t\in \Bbb N$ we have $$\{\tau \leq t\} = \bigcup\limits_{n=1}^t \{X_n \in B\} \in \sigma(X_1,\ldots,X_t) =: \mathcal F_t$$

So $\tau$ is a stopping time.