Is the supremum of a Brownian motion almost surely finite?

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!

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