Use the Borel-Cantelli Lemma and reflection principle to show that $$\limsup_{t\to \infty}B_t=\infty$$
The proof starts by constructing the following events $$E_i:=\{\inf\{t\ge0:B_t=i\}\lt \infty\}$$.
$$\mathbb P(E_i)=\lim_{t\to \infty} \mathbb P(\max_{s\le t}B_s\ge i)$$ $$=\lim_{t\to \infty} 2\mathbb P(B_t\ge i)$$ $$=\lim_{t\to \infty} 2\mathbb P(B_1\ge \frac{i}{\sqrt t})=1$$
Then later in the proof, it is claimed that $E_i$ are independent for any $i$. I cannot figure out why. Any help?
Then $$\sum_{i=1}^\infty \mathbb P(E_i)=\infty$$ by BC we have the result.
Trivial events (like the $E_i$) are always independent: $\Bbb P[E_i\cap E_j]=1= 1\cdot 1=\Bbb P[E_i]\cdot\Bbb P[E_j]$.