Let $f_n,f\in C_c(\mathbb{R}^n)$, and $f_n \rightarrow f$ uniformly on $\mathbb{R}^n$. If $g$ goes to infinity at zero but is otherwise continuous, assuming $f_ng$ is integrable, do we have the following convergence?
$$\int_{\mathbb{R}^n} f_n g \longrightarrow \int_{\mathbb{R^n}} fg $$
Clearly on a compact set not including zero, $g$ is bounded, so $f_ng$ converges uniformly to $fg$ on that set, meaning the integrals converge on that set, but since $g$ blows up to infinity, I don't think $f_ng$ converges to $fg$ on $\mathbb{R}^n$.
I'm having trouble seeing if the integral convergence holds over the whole of $\mathbb{R}^n$. I tried splitting it up into rings of compact sets, but then I ended up with an infinite sum of integrals all converging, but got nowhere with this.