Why does $1 \leq C\sup \limits_{0\leq t \leq 1}( |B_t|)$ P -as where $B$ is the standard B.M for some $C>0$ does not hold ?
I am trying to show by contradiction that the Burkholder-Gundy inequalities do not extend to random times. In order to prove this I need to show the above .
I am not so sure how to proceed. I know that since Brownian motion is almost surely continuous , $|B_t|$ is almost surely bounded on the interval $[0,1]$ and hence $\sup \limits_{0 \leq t \leq 1} |B_t| \leq C'$ almost surely. I can't argue why however high $C$ is there is always a set of non-zero measure such that on that set we have