Let $f_n$ be a continuous function sequence, $\lim_{n\to\infty}f_n(x)=0$. Can we show that for any $\phi\in C^\infty(\Bbb R)$ with compact support, $\int_{\Bbb R} f_n(x)\phi(x)dx\to 0\ (n\to\infty)?$
I could not provide an example, or prove it.
I think it is not true, but I do not have an example.
(I know that if $f_n$ is uniformly bounded, then it is true by Lebesgue's dominated theorem)
This is incorrect.
Consider $f(x) = xe^{-x}$ for $x > 0$, $f(x) = 0$ for $x \le 0$, and set $f_n(x) = nf(nx)$. Then $f_n(x) \to 0$ as $n \to \infty$ for all $x$, but $\int_{\mathbb{R}}\phi(x) f_n(x) \to \phi(0)$ for all $\phi \in C^\infty_0$.