Uniform convergence on compact subset and integration.

Question

Let $(a,b)$ be a open interval and $f_n$ be a sequence of continuous functions on $(a,b)$. Given $f_n$ is uniformly convergent to $0$-function on any compact subset of $(a,b)$. I want to show that: $$\lim_{n\to\infty} \int^a_b f_n=0$$ I know that by uniform convergence I can switch the integration and limit, so the intergration is 0 for any compact subset of $(a,b)$. But I can't pass the conclusion through the whole $(a,b)$. This seems to be intuitive but I cannot write down a rigirous proof of it. Any help is apperetiated.

Answer 1

This is false. Let $a=0,b=1,f_n(x)=n-n^{2}x$ for $0<x<\frac 1 n$ and $f_n(x)=0$ for $x\geq \frac 1 n$. Can you check that this serves as a counterexample?