Maximum of stopping times is not a stopping time

Question

Is the maximum of two stopping times a stopping time? I wrote

$$\{max(\tau, \sigma) \le t\} = \{\tau \le t\} \cap \{\sigma \le t\} \in \mathcal F_t$$

Because both $\tau$ and $\sigma$ are stopping times. In my book there is written that it isn't though. Can you tell me where the error is?

Answer 1

I have read the book solution incorrectly. Indeed the maximum of two stopping times is a stopping time, as I showed in the question.