I am reading the proof of the Approximation theorem 6.5 from Michael Steele's Stochastic Calculus and Financial Applications, and there is one particular point in the proof that has me stuck.
Specifically towards the end of the proof, the author claims that for any $f \in \mathcal H^2$, and any $\epsilon >0$ there is $f_0 \in \mathcal H^2$ that is bounded such that $\lVert f-f_0 \rVert_2\leq\epsilon$.
Relevant notation:
The space $\mathcal H^2$ consists of all measurable adapted processes $\{f_t\}_{t \geq 0}$ (with respect to the standard Brownian filtration) such that
$$\mathbb E\left[\int_0^T f^2_u du\right] < \infty$$
Can someone give some hint or reference why this is true?
For a process $f= (f_t)_{t\in [0,T]}\in\mathcal{H}^2$ define for every $n\in\mathbb{N}$ a bounded process $f^n= (f_t^n)_{t\in [0,T]}\in\mathcal{H}^2$ via $$ f^n_t(\omega) := f_t(\omega) {\bf 1}_{[-n,n]}(f_t(\omega)). $$ Here, ${\bf 1}_A$ is the function that is equal to $1$ on $A$ and $0$ else. On the one hand, one has for every $\omega\in\Omega$ and $t\in [0,T]$ $$ (f_t(\omega) - f^n_t(\omega))^2 = f_t(\omega)^2 \underbrace{{\bf 1}_{[-n,n]^c}(f_t(\omega))}_{\to 0} \to 0 $$ as $n\to\infty$, on the other hand $(f_t(\omega) - f^n_t(\omega))^2\le f_t(\omega)^2$. Hence, Fubini's theorem and dominated convergence theorem implies $$ \|f- f^n\|_2 = \int \int_0^T (f_n(\omega) -f_2^n(\omega))^2 dt d{\mathbb P}(\omega) \to 0. $$