I have a question:
Given two disjoint intervals $[a,b]$ and $[c,d]$, how to prove almost surely we have
$$\sup_{t\in[a,b]}B_t\neq\sup_{t\in[c,d]}B_s$$
where $B$ is a standard brownian motion. I have no idea about this problem. Does someone have an idea? Thansk a lot!