In some results on Hölder continuity with regards to standard Brownian motion, the following is asserted without proof.

It is not hard to see that for every $k < \infty$, and every $\epsilon > 0$,$$\mathbb{P}\left\{ \sup_{0 < s < t < 1} {{|B_t - B_s|}\over{\sqrt{t - s}}} \ge k\right\} > 1 - \epsilon,$$where $B_t$ is a standard Brownian motion.

To me, this is not all that trivial. Could anyone explain why this is the case?

**EDIT:** Sorry, I corrected my mistake.