I'm studying about the reflection principle of the brownian motion, and I found that this result is a direct consequence of this principle:
Let $B_t$ a brownian motion, then for every $a \in \mathbb{R} \ $,
$$\mathbb{P}(\lim_{t \to \infty} \sup_{s\in [0,t]} B_s > a) = 1$$
I'm trying to prove this statement using the reflection principle but I'm totally lost. I can't see how are those results related.