I quote Øksendal (2003).
Itô integral. Let $\mathcal{V}=\mathcal{V}(S,T)$ be the class of functions $f(t,\omega):[0,\infty)\times\Omega\to\mathbb{R}$ such that $(t,\omega)\to f(t,\omega)$ is $\mathcal{B}\times\mathcal{F}$-measurable (where $\mathcal{B}$ denotes the Borel $\sigma$-algebra on $[0,\infty)$), $f(t,\omega)$ is $\mathcal{F}_t$-adapted and $\mathbb{E}\bigg[\int_{S}^T f(t,\omega)^2 dt\bigg]<\infty$.
[...] For functions $f\in\mathcal{V}$ we will now show how to define the Itô integral $$\mathcal{I}[f](\omega)=\int_{S}^{T}f(t,\omega)dB_t(\omega)$$ where $B_t$ is $1-$dimensional Brownian motion.
[...] The idea is natual: First we define $\mathcal{I}[\phi]$ for a simple class of functions $\phi$. Then, we show that each $f\in\mathcal{V}$ can be approximated by such $\phi$'s and we use this to define $\int fdB$ as the limit of $\int\phi dB$ as $\phi\to f$.
Recall that a function $\phi\in\mathcal{V}$ is called elementary if it has the form $$\phi(t,\omega)=\sum_j e_j(\omega)\cdot\chi_{[t_j, t_{j+1}]}(t)$$ Lemma (Itô isometry). If $\phi(t,\omega)$ is bounded and elementary then $$\mathbb{E}\bigg[\bigg(\int_{S}^{T}\phi(t,\omega)dB_t(\omega)\bigg)^2\bigg]=\mathbb{E}\bigg[\int_{S}^T\phi(t,\omega)^2dt\bigg]\tag{1}$$ [...] The idea is now to use the isometry $(1)$ to extend the definition from elementary functions to functions in $\mathcal{V}$. We do this in several steps:
STEP 1. Let $g\in\mathcal{V}$ be bounded and $g(\cdot,\omega)$ continuous for each $\omega$. Then there exists elementary functions $\phi_n\in\mathcal{V}$ such that $$\mathbb{E}\bigg[\int_S^T(g-\phi_n)^2dt\bigg]\to 0\text{ as }n\to\infty\tag{2}$$;
STEP 2. Let $h\in\mathcal{V}$ be bounded. Then there exist bounded functions $g_n\in\mathcal{V}$ such that $g_n(\cdot,\omega)$ is continuous for all $\omega$ and $n$, and $$\mathbb{E}\bigg[\int_S^T(h-g_n)^2dt\bigg]\to0\tag{3}$$;
STEP 3. Let $f\in\mathcal{V}$. Then there exists a sequence $\{h_n\}\subset\mathcal{V}$ such that $h_n$ is bounded for each $n$ and $$\mathbb{E}\bigg[\int_S^T(f-h_n)^2dt\bigg]\to0\text{ as }n\to\infty\tag{4}$$ That completes the approximation procedure. [...] If $f\in\mathcal{V}$ we choose, by STEPS 1-3, elementary functions $\phi_n\in\mathcal{V}$ such that $$\mathbb{E}\bigg[\int_S^T|f-\phi_n|^2dt\bigg]\to0\tag{5}$$Then define $$\mathcal{I}[f](\omega)=\int_{S}^{T}f(t,\omega)dB_t(\omega)=\lim\limits_{n\to\infty}\int_S^T\phi_n(t,\omega)dB_t(\omega)\tag{6}$$
My question is:
why are those three STEPS necessary so as to get to $(5)$? In other words, which is the rationale of each of those three STEPS? Why has one to follow each of them so as to get to state that each $f\in\mathcal{V}$ can be approximated by such $\phi$'s?
Could you please detail your answer step-by-step?