I'm trying to figure out a statement from page 166 of the book "Brownian Motion: An introduction to stochastic processes" by Schilling, Partzsch (2012). The chapters illustrates how fast the brownian paths grow as $t \rightarrow \infty$.
What I don't understand is the following statement:
"A first estimate (of the growth of the brownian paths) follows from the strong law of large numbers. Write for $n \geq1$
and observe that the increments of a brownian motion are stationary and independent random variables (with B(t) being a one dimensional standard brownian motion). By the strong law of large numbers we get that
$$B(t,\omega) \leq \epsilon t$$
$\forall \epsilon>0$ and all $t \geq t_0(\epsilon , \omega)$"
How can I use SLLN to obtain such bound?
Thanks in advance.