I have a problem proving the following statement.
Consider $T>0$ and $t_0=0<t_1<t_2<...<t_n$ a partition of $[0,T]$ such that the norm of the partition tend to $0$ when n goes to infinity (i.e. $max_j (t_{j+1}-t_{j}) \rightarrow 0$ when n goes to infinity).
I would like to prove that, in this case, $max_j|W_{t_{j+1}}-W_{t_{j}} |\rightarrow 0$ when n goes to infinity, where $W_{t_{j}}$ are Brownian motions and the convergence is at least in probability.
I was thinking about using the almost sure continuity of brownian motion to conclude but as the number of increments increases as n increases, I am not sure it is enough to conclude.
Thank you in advance!