Let $(B_t)$ be a Brownian motion. Consider the event : $B(n)>a \sqrt n $occuring infinitely often.
I want to prove that this event has probability 1. we can see that, by rescaling property, $$\mathbb{P}(B(n)>a\sqrt{n} \mbox{ occurs infinitely often})\geq \limsup_{n\rightarrow\infty}\mathbb{P}(B(n)>a\sqrt{n})=P(B(1)>a)>0$$
Now I want to see that this is a 0-1 event to conclude using Blumenthal 0-1 law
Hint: use the first Borel-Cantelli lemma
I feel sorry about the above hint, as I thought this is a typical question using B-C lemma, but actually it does not work here.
You are on the right track, and it remains to be shown this is an tail event, which is not difficult. In fact, this is the Theorem 8.2.8 in Durrett's book-- Probability: Theory and Examples (4th edition).