What I want to show is that $$\int_0^t\phi^n_t \, dB_s = \sum_{i=1}^{K_n-1}B_{t_{i-1}^n}(B_{t_{i}^n}-B_{t_{i-1}^n}) \to \int_0^tB_s\,dB_s$$ where $\phi^n_t=\sum_{i=1}^{K_n-1}B_{t_{i-1}^n} \mathbb1_{(t_{i-1}^n, t_i^n)}(t)$.
For doing so is sufficient to prove that $$E\bigg[\int_0^t(\phi^n_t-B_s)^2\,ds\bigg] = \|\phi^n_t-B_s\|_{\lambda^2}\to0$$
Then $$\int_0^t\phi^n_t\,dB_s\to\int_0^tB_s\,dB_s$$ in $L^2$ follows by the construction of the stochastic integral.
However this is the construction followed starting from a simple process, but in this case $\phi^n_t=\sum_{i=1}^{K_n-1}B_{t_{i-1}^n} \mathbb1_{(t_{i-1}^n, t_i^n)}(t)$ is it a simple process? For being a simple process $B_{t_{i-1}^n}$ should be a bounded random variable. Is it the case? It seems to me that depend also on time (and so is not a random variable but a stochastic process) and it also seems to me that is not bounded since a B.M reaches every point an infinite number of times. And finally, if it is not a simple process why this construction still hold?