Okay, I have cleared up some of my proof. Sorry for confusion (it was first time to be in this StackExchange, to make an excuse).
HINT(given): Use the independent increments property of brownian motion and continuity property.
What I want to do is to show that probability of having each random moment of Brownian motion smaller than $\epsilon$ is zero.
$P(\sup (B(t_2) - B(t_1) < \epsilon), sup (B(t_3) - B(t_2)) < \epsilon, ..., ) = \prod P (sup (B(t_i ) - B(t_i-1)) < \epsilon), $
by independent increment property.
I think I can use the continuity property to show that the product of such is equal to zero, and then probability of having each Brownian smaller than $\epsilon$ is zero. Thus, $\limsup \left| B(n) \right| = \infty.$
I want to know:
i) Am I on the right track?
ii) How can I move to the rest of the proof?
Thanks in advance.