Let $\tau$ be a stopping time w.r.t a filtration $(F_n)_n$ and $(X_n)_n$ a discrete time martingale w.r.t the same filtration.
Usually in the exercises, in order to apply the optional stopping thm, we have to show that $$P(\tau < \infty)=1$$
Then, we are allowed to write $X_{n \wedge \tau} \rightarrow X_{\tau}$ a.s.
Indeed, since $\tau$ is finite, as $n$ gets larger and larger we have that $\exists N \in \mathbb{N}: \forall n > N: \tau< n$ and hence $$n \wedge \tau \rightarrow \tau$$
My questions then is: if $\tau = \infty$ a.s., why doesn't that hold? I mean, I can't figure out what could be the value of $lim_{n} (n \wedge \tau)$