Let $A_t$ be a standard Brownian motion. Where can I find a reference to/can anybody supply a proof of the fact that with probability $1$ there exists a sequence of partitions $\{t_{k, n} : k = 1, \dots, k_n\}$ $$0 = t_{0, n} < t_{1, n} < \dots < t_{k_n, n} = 1,$$with$$\lim_{n \to \infty} \max \{t_{j, n} - t_{j - 1, n} : j = 1, \dots, k_n\} = 0$$and$$\liminf_{ n \to \infty} \sum_{j=1}^{k_n} (A(t_{j, n}) - A(t_{j-1, n}))^2 > 1?$$All the sources I have been looking at have just been assuming the existence of this sequence? It's not very clear to me how this is clear.
Thanks in advance!
Have a look here: S. J. Taylor, "Exact asymptotic estimates of Brownian path variation", Duke Math. J. vol. 39 (1972), pp. 219–241.