Let $f_n : [0, \infty) \rightarrow \mathbb{R}$ be a sequence of bounded functions, which uniformly converge to $f: [0,\infty) \rightarrow \mathbb{R}$ Also assume $f_n|[a,b]$ is Riemann-Integrable $\in R([a,b])$ und $\int_0^{\infty} f_n(x)dx$ exists.
Now it has to be shown whether $\int_0^{\infty}f_n(x)e^{-x}dx \rightarrow \int_0^{\infty} f(x)e^{-x}dx \quad n \rightarrow \infty$ holds or not.
I was already able to show that $\int_0^{\infty}f_n(x)dx \rightarrow \int_0^{\infty} f(x)dx \quad n \rightarrow \infty$ does not hold in general.
But I have the feeling it is different if we include $e^{-x}$ into our function. Because $f_n$ is bounded, we can find $x_0$ such that $f_n \leq e^{x_0}$ and therefore $f_ne^{-x} \rightarrow 0$ as $x \rightarrow \infty$. I have tried to consider $\int_0^{x_0}f_n(x)e^{-x}dx + \int_{x_0}^{\infty}f_n(x)e^{-x}dx$ but got stuck and also considered $\int_0^{\infty}e^{-x} (f_n(x)-f(x)) dx$ and could not proceed. I would value any input.