When I have shown, for $s\le t$ and for two continuous stochastic process an inequality:
$ X_s \le Y_t$ P-a.s.
How can I deduce that this P-a.s. simultaneously for all rational $s\le t$ ? Thank you for your help
EDIT: According to Ilya's answer, I see that we have $P(X_s\le Y_t\text{ simultaneously for all rationals }s\le t) = 1.$ How could we use continuity of $X,Y$ to deduce $P(X_s\le Y_t,s\le t)=1$. Of course we take sequences of rational, however I mess up the details. So a detailed answer how to do this, would be appreciated.