I am trying to prove the unboundedness of variations in a Brownian motion in the following way: Consider the interval $[0,1]$ and let $B$ be the standard Brownian motion. We consider the sub-intervals $[\frac{i}{n}, \frac{i+1}{n}]$. The goal is to show that the summation $\sum_{i=1}^n |B(\frac{i}{n}) - B(\frac{i+1}{n}) |$ is almost surely unbounded. I greatly appreciate any comment or hint to prove this.
My attempt: I was thinking on finding a lower bound for $|B(\frac{i}{n}) - B(\frac{i+1}{n}) |$ and show that this lower bound goes to $\infty$. We know $B(\frac{i}{n}) - B(\frac{i+1}{n}) \sim Normal(0,1/n)$. I was wondering if this is a good way to prove the above claim and how I can find a lower bound. Thank you.