Quadratic Variation of a Brownian motion $B$ over the interval $[0,t]$ is defined as the limit in probability of any sequence of partitions $\Pi_n([0,t])=\{0=t^n_0<\cdots
And for any such sequence of partition we have then $[B]_t=P-\lim_{n\to \infty} V([0,t],\Pi_n)(B_.)=t$.
Nevertheless when you take the sup over all finite partitions of $[0,t]$ then it is a known fact that almost surely $\sup_{\Pi\in \mathrm{partition}([0,t])} V([0,t],\Pi)(B_.)=+\infty$.
I have never been able to derive this fact properly and in every details.
I'd be really gratefull if anyone could take the time to provide a detailed proof of this fact.