I found the following claim interspersed in a text on the growth of Browinan paths.
"By the strong law of large numbers we get that $$ B(t, \omega) \le \epsilon t $$ for any $\epsilon > 0$ and all $t \ge t_0(\epsilon, \omega)$."
I can see the following: given any $\epsilon>0$, pick any $t_1 > 0$, then by the strong law of large numbers we have that $$ \frac{B(nt_1)}{n} = \frac{1}{n}\sum_{j=1}^n \left[ B(jt_1) - B((j-1)t_1) \right] \to 0 \text{ a.s.} $$ Thus for any particular $\omega$, chose $n_\epsilon$ such that $$ \left|\frac{B(nt_1, \omega)}{n} - 0 \right | \le \epsilon, \text{ for } n \ge n_\epsilon. $$ In particular $$ B(nt_1, \omega) \le \epsilon n, \text{ for } n \ge n_\epsilon. $$
But is it possible to let $t_0(\epsilon, \omega) := n_\epsilon t_1$ and conclude that $$ B(t, \omega) \le \epsilon t \text{ for } t \ge t_0 \text{ ?} $$
