I am studying chapter 2 of the book "Stochastic Differential Equations and Diffusion Processes by Nobuyuki Ikeda and Shinzo Watanabe." I am struggling to write down proof for the following result which is in Proposition 1.1 (page number 49).
The stochastic integral with respect to an $(\mathscr F_t)$-Brownian motion has the following property: For each $t \gt s \ge 0, E[(I(\phi)(t) - I(\phi)(s))^2 | \mathscr F_s] = E[\int_{s}^{t} \phi^2(u, \omega)du | \mathscr F_s] \; \; \; \text{a.s.}$
I have shown the above equality for $\phi \in \mathscr L_0.$ I wish to show this equality for $\phi \in \mathscr L_2$.
I want to use the following result which is already proven in the book:
Let $\phi \in \mathscr L_2.$ There exists a sequence $\phi_n \in \mathscr L_0$ such that $|| \phi_n - \phi||_2 \to 0$; which implies $||I(\phi_n) - I(\phi)|| \to 0.$
Considering the LHS first, I want to show that $E[(I(\phi_n)(t) - I(\phi_n)(s))^2 | \mathscr F_s] \to E[(I(\phi)(t) - I(\phi)(s))^2 | \mathscr F_s]$ as $n \to \infty \; \text{a.s.}$ In order to use the conditional dominated convergence theorem for this purpose, it needs to be shown that $I(\phi_n)(t) - I(\phi_n)(s) \to I(\phi)(t) - I(\phi)(s) \; \text{a.s.}$ and $|I(\phi_n)(t)-I(\phi_n(s))| \le Y \; \forall n$ for some Y integrable.
Now $||I(\phi_n) - I(\phi)|| \to 0 \Rightarrow ||I(\phi_n) - I(\phi)||_t \to 0 \; \forall t \gt 0 \Rightarrow E[(I(\phi_n)(t) - I(\phi)(t))^2]^{\frac 12} \to 0 \; \forall t \gt 0.$ But I am not sure how to proceed from here. Also, can someone give an outline for the RHS part?
Ikeda, Nobuyuki; Watanabe, Shinzo, Stochastic differential equations and diffusion processes., North-Holland Mathematical Library, 24. Amsterdam etc.: North-Holland; Tokyo: Kodansha Ltd. xvi, 555 p. {$} 147.25; Dfl. 280.00 (1989). ZBL0684.60040.