Let $\{t_i\}_{i=1}^n$ be a partition of $[0,t]$ and $W$ a standard Brownian motion. Write $W_i$ for $W_{t_i}$.
Show $ \lim \sum W_{i} (W_{i+1}-W_i)=\frac12 W^2_t-\frac12 t $ where the limit is in probability.
The proof is in our textbook (Kurtz, Stochastic Analysis). It goes as follow \begin{align} \lim \sum_{i=1}^n W_{i} (W_{i+1}-W_i) &= \lim \sum_{i=1}^n \left( W_i W_{i+1} - \frac12 W^2_{i+1}-\frac12 W^2_i \right)+\sum_{i=1}^n \left( \frac12 W^2_{i+1} - \frac12 W^2_i \right) \\ &=\frac12 W_t^2 - \lim \frac12 \sum_{i=1}^n \left( W_{i+1}-W_{i} \right)^2 \\ &=\frac12 W_t^2 - \frac12 t^2 \end{align}
How does the second equality follows?