Derivation — Quadratic Variation of Brownian Motion

This derivation sub-note carries the $L^2$ computation underpinning Theorem 2.8.2 from the main lesson: that the partition sums $Q(t; \Pi_n) = \sum_j (W_{t_j} - W_{t_{j-1}})^2$ converge in probability to $t$ (and a.s. under the summability condition). The main lesson Quadratic Variation and Non-Differentiable Paths gives the interpretation; here we do the algebra.