Let $B$ be a Brownian motion, I want to show that $\lim_{t\rightarrow \infty} \frac{B_t}{t}=0$.
I first tried to prove the statement in the case where the index set are the natural numbers. Indeed since $B_{n+1}-B_n\sim \mathcal{N}(0,1)$ i.i.d. we get by the strong law of large numbers that $$\frac{B_n}{n}=\frac{1}{n}\sum_{k=0}^{n-1} (B_{k+1}-B_k)\rightarrow \Bbb{E}(B_2-B_1)=0$$ But now I would like to extend this to all $t> 0$. I thought about approximating any strictly positive real number by a sequence of rational numbers, but I'm still a bit unsure how to do that because I would also need to show the statement for rational numbers.
Can someone help me?