So I'm trying to read through Stochastic Differential Equations: An Introduction with Applications by Oksendal.
When he defines the integral with respect to brownian motion in chapter 3, he defines $\mathcal{V} = \{f: f \text{ is }\mathcal{B\times \mathcal{F}} \text{ measurable, } f \text{ is } (\mathcal{F}_t)_{t\geq0} \text{ adapted and } \mathbb{E}[\int_{(S,T)}f^2dt] < \infty \}$ and he says that if $f \in \mathcal{V}$, we choose a sequence of elementary functions $(\phi_n)_{n \in \mathbb{N}}$ such that $$\mathbb{E}\bigg[\int_{(S,T)}(f-\phi)^2 dt\bigg] \rightarrow 0$$ then we define $$\int_{(S,T)}fdB_s = \lim_{n \rightarrow \infty}\int_{(S,T)}\phi_ndB_s \hspace{.5cm} \text{(in $L^2$)}$$
He says the limit exists because the sequence of integrals of elementary functions with respect to brownian motion is cauchy by the ito isometry. This is what I am confused about. I believe my question boils down to:
Does $$\vert \vert f \vert \vert = \mathbb{E}\bigg[\int_{(S,T)}f^2dt \bigg]^{1/2}$$ satisfy the triangle inequality on $\mathcal{V}$? If so, how can I show this?