Inf is not a stopping time in general If ${\tau_n}$ , $n=1,2,3...$ are stopping times to a given filtration $F_t$, why in general it's not true to claim that $\inf_n {\tau_n}$ is a stopping time also?
Thanks

I guess we are talking about continuous time: stochastic basis $(\mathcal F_t)_{t \in (0,\infty)}$, say.  
Write $\sigma = \inf_n \tau_n$.  Can we verify that $\{\sigma \le 5\} \in \mathcal F_5$ ???
Well, suppose $B \in \bigcap_{t>5}\mathcal F_t$ but $B \not\in \mathcal F_5$.  Then for each $n$,
$$
\tau_n(\omega) = \begin{cases}
5+\frac{1}{n},\qquad \omega \in B
\\
17,\qquad \omega\not\in B
\end{cases}
$$
defines a stopping time, but
$$
\sigma(\omega) = \begin{cases}
5,\qquad\omega \in B
\\
17,\qquad\omega\not\in B
\end{cases}$$
is not a stopping time.
