Max of Brownian motion with drift is finite almost surely For $B_t$ Brownian Motion with drift $\mu&lt;0$, I need to prove that the max value, $X = \max_{0&lt;t&lt;\infty}B_t$ is finite almost surely, ie $P(X&lt;\infty)=1$.

Now, I know that because the mean is negative, it will go more and more negative, and it is also a supermartingale. But I don't know how to prove almost surely...

Appreciate any hints.
Hint: Try the strong law of large numbers.  What does it say about $\lim_{t \to \infty} B_t/t$?  What does this say about the sign of $B_t$ for large $t$?
Actually that is wrong in the sense that it is not finite in the limit.  Brownian Motion with a negative drift will wander off to -∞ almost surely. It will tend to go down in a linear fashion with the slope equal to the drift parameter.