Let $\{f_{n}\}_{n=1}^{\infty}$ be a sequence of continuous functions on $\mathbb R$ , $f_n\geq 0$ for all $n$, where $|f_{n}(x)|\leq A$ for all $n=1,2,3,...$ on $\mathbb R$. Assume that $\lim_{n\to \infty}f_{n}(x)=0$ uniformly on $\mathbb R$, and $\int_{-\infty}^{\infty}|f_{n}(x)|dx<\infty$ for all $n=1,2,3,...$ . Is it true that $ \lim_{n\to \infty}\int_{-\infty}^{\infty}|f_{n}(x)|dx=\int_{-\infty}^{\infty}\lim_{n\to \infty}|f_{n}(x)|dx=0 $
The Dominated convergence, Monotone, and Fatou's theorems doesn't apply here. Any help?