suppose we have a sequence of continuous functions $(f_n)_{n\geq 1}$ with $f_n:[0,\infty)\rightarrow \mathbb{R}$ such that $\int_{0}^{\infty}\vert f_n(x) \vert<\infty$ for all $n\in\mathbb{R}$. Further, let $\lim\limits_{n\rightarrow \infty}f_n(x) = 0$ for all $x\geq 0$.
Is it true that $$\lim\limits_{n\rightarrow \infty}\int_{0}^{\infty}f_n(x)\cdot g(x) dx = 0,$$
if $g:[0,\infty)\rightarrow \mathbb{R}$ is a continuous and absolutely integrable function (i.e. $\int_{0}^{\infty}\vert g(x) \vert<\infty$)? Can I interchange the order of integration and the limit? (If needed, we may also assume that $\int_{0}^{\infty}\vert f_n(x)\cdot g(x) \vert<\infty$ for all $n$).
Best regards