I understand almost all of the proof except for this one part. I proceed to break up the Stratonovich sum as follows:
$$\begin{align} \displaystyle \sum B_{\frac{t_j+t_{j+1}}{2}}(B_{t_{j+1}}-B_{t_{j}}) &= \\ \sum [ B_{\frac{t_j+t_{j+1}}{2}}(B_{t_{j+1}}-B_{\frac{t_j+t_{j+1}}{2}})+B_{t_{j}}(B_{\frac{t_j+t_{j+1}}{2}}-B_{t_j})]+ \sum (B_{\frac{t_j+t_{j+1}}{2}}-B_{t_j})^2 \end{align}$$
In my book, it says, "Clearly, the first sum converges to $$\int^{T}_{0} B_t dB_t $$.
Why is this 'clear'? I can sort of heuristically reason this result but it's not apparent to me why this is obvious? Is there a more rigorous way to see this?