I am struggling with the following comment in one of the books I am reading:
For example, if $f_m \in C(\mathbb{R}^n) ,m\in \mathbb{N}$, is a sequence of functions such that it converges uniformly to $f \in C(\mathbb{R}^n)$ on any compact subset $K \subset \mathbb{R}^n$ and $\phi \in C^{\infty}_c$ then $lim_{m \rightarrow \infty} \int_{R^n} f_m(x)\phi(x)dx=\int_{R^n} f(x) \phi(x)dx$
I can't really see why I can interchange the limit and the Integral. My Problem is the "it converges uniformly ... on any compact subset".
It would make sense if I would know that $f_m$ has compact support. I hope someone could tell what I am overlooking.