Is the supremum of a Brownian motion almost surely finite?

Question

I have a Brownian motion $\{B(t) : t \in [0,1] \}$ and I'm trying to figure out if

$$ \sup \limits_{t \in [0,1]} \vert B(t) \vert < \infty $$

almost surely holds. Does this immedeately follow as $[0,1]$ is a compact interval and the paths $t \mapsto B(t)$ are almost surely continous, such that the extreme value theorem is applicable?

Thanks!

Answer 1

Surely. For any $\omega$ the supremum is finite because continuous functions on a compact interval are bounded