For this assignment I'm working on, I was able to prove that:
$$\limsup_{t \rightarrow \infty} \frac{B_t}{\sqrt t} = \infty$$
where $B_t$ is a Brownian Motion. I'd like to be able to prove:
$$\liminf_{t \rightarrow \infty} \frac{B_t}{\sqrt t} = -\infty$$
I'm wondering if there's a quick way to get that result from the $\limsup$ result by symmetry. I'm sure I could get the result the same way I got the first one with the Kolmogorov 0-1 Law and using discrete time, but there should be a way to use a symmetry kind of argument. I'm just struggling a bit to do it formally instead of saying "The result follows by symmetry"