Let $(B_t)_{t \geq 0}$ be a Brownian motion. The objective is to prove that \begin{align*} \limsup_{t \to \infty} \frac{B_t}{\sqrt{t}} = \infty. \end{align*} By the scaling property of Brownian motions. It can be shown for $K \in (0,\infty)$ that \begin{align*} & P( B_n > K \sqrt{n}\ \ \text{ i.o.}) \\ &\geq P(B_1 > K ) \\ &>0. \end{align*} Now, I want to work towards the result saying that $B_n = \sum_{k=1}^n X_k$ where $X_k = B_k - B_{k-1}$ of independent random variables and using subsequently the Kolmogorov 0-1 Law.
However for me it is unclear how to use the Kolmogorov 0-1 Law to obtain a satisfactory result. For me the Kolmogorov 0-1 Law is about a sequence of independent $\sigma$-algebras $(F_n)_{n \geq 0} \in \mathcal{F}$. Then for any event $F \in \bigcup_{n=1}^\infty G_n$ one has either $P(F)=0$ or $P(F)=1$, where $G_n = \sigma(\bigcup_{k=n}^{\infty} F_k)$. How to apply this theorem?