My question is, as stated in the title: how do I prove that if $f,g$ are Riemann integrable over the reals ($\mathcal{R}(\mathbb{R})$), then $f \cdot g \in \mathcal{R}(\mathbb{R})$. However, I have been asked to prove this by using the definition given in class, which is different than the standard definition one would encounter in most analysis courses/texts.
Note: Let $\mathcal{S}(\mathbb{R})$ represent the simple functions. Thus, if $f \in \mathcal{S}(\mathbb{R})$, then $f = \alpha_1 \chi_{I_1}+ ... +\alpha_n \chi_{I_n}$, where $I_1,...,I_n$ are the bounded intervals over $[a,b]$. Also given was (not sure if it is necessary):
$\int f = \alpha_1 \lambda(I_1)+...+\alpha_n \lambda(I_n)$, where $\lambda(I)=b-a$.
The definition is as follows:
$f \in \mathcal{R}(\mathbb{R})$ if for all $\varepsilon>0$ there exist $g,h \in \mathcal{S}(\mathbb{R})$ such that $$g_{\varepsilon}\leq f \leq h_{\varepsilon} $$ and $$\int (h_{\varepsilon}-g_{\varepsilon})<\varepsilon$$
I have no idea where to start. If additional information is needed to answer the question, please let me know. I don't entirely understand the definition so it is difficult to know whether I have provided enough info. Any help is appreciated.