Product of integrable by bounded function (Daniell approach)

Question

Define the space $L^1(\mathbb{R})$ as the completion, in the norm $\Vert\cdot\Vert_1$, of the space of continuous functions with compact support $\mathcal{C}_0(\mathbb{R})$. If $f$ is integrable and $g$ bounded integrable, it is known that $fg$ is integrable (for instance, the product is measurable and bounded by the integrable function $M|f|$, where $M>0$ is a bound for $g$). But I want to prove that using Cauchy sequences in $\mathcal{C}_0(\mathbb{R})$. There are Cauchy sequences $(f_n),(g_n)\subset\mathcal{C}_0(\mathbb{R})$ converging in $\Vert\cdot\Vert_1$ to $f,g$, so one could try the obvious sequence $(f_ng_n)$ and see if it converges to $fg$. However, the usual trick of adding and substracting leads me to a term $ \int|g_n-g_n||f_n|$ which I do not know how to deal with. Any idea on how to proceed?