I have a process $(B_t(\omega))$ with finite quadratic variation:
$E[\int_{0}^{T}B_t^2dt] < \infty $ (1)
According to my material, I can approximate the integral $\int_{0}^{T}B_tdB_t$ by a simple process $H_{t}^{n}$. Thereby, I build on the following notion of convergence:
$lim \ (as \ n \rightarrow \infty) \int_{0}^{T} | H_{t}^{n}-B_{t}|^2 dt = 0$ (2)
I understand how I can apply (2) to solve my integral (i.e., from a "mechanical" point of view). However, I struggle to see any connection between (1) and (2). Does (2) only hold if (1) holds? If yes, why?