Why $ \lim_{t\to\infty} \frac{W_t}{t} = 0 $
I tried to write it as Wiener process is defined which is:
$ \lim_{t\to\infty} \frac{W_t}{t} = \lim_{t\to\infty}\lim_{N\to\infty} \frac{\sum_{k=1}^{\lfloor Nt \rfloor} \varepsilon_{k}}{\sqrt{N}*t}*\frac{\lfloor Nt \rfloor}{\lfloor Nt \rfloor} $
and then use a Law of large numbers but I got expression $ 0*\infty $
any hints or solution would be appreciated.
Ok, I tried different approach, we know that, $$ Z_t=t*W_\frac{1}{t} $$ is also a wiener process, then wanted limit is equiwalent to $$ \lim_{t\to 0 } \frac{Z_\frac{1}{t}}{\frac{1}{t}}=\lim_{t\to 0} t*Z_\frac{1}{t} $$ which is 0 , is it true ?