Signs of a sequence of Brownian motion

Question

I am reading some course materials which say $$ \mathbb{P}(n \in \mathbb{N}: B_{1/n}>0 \text{ infinitely often}) \geq \frac{1}{2}, $$ then by Blumenthal's 0-1 law...

May I ask about where this $1/2$ comes from?

Answer 1

$$ \mathsf{P}(B_{1/n} >0 \text{ i.o.})\ge \limsup_{n\to\infty}\mathsf{P}(B_{1/n}>0)=\frac{1}{2}. $$