Let {${g_n}$} be a sequence of measurable functions defined on $[a,b]$.Suppose $|g_n(x)|\leq K$ for a.e $x$, where $K< \infty $. Suppose further $\lim_{n\to\infty}\int_{a}^{c} g_n(x)dx=0$ for all $a \leq c\leq b$.
Prove that for any $f \in \mathbb{L^1{[a,b]}}$, $\lim_{n\to\infty}\int_{a}^{b} f(x)g_n(x)dx=0$.
My thoughts:The idea is to use DCT for $f_n=fg_n$ on $[a,b]$. Since $g_n$ is int'ble and $f \in \mathbb{L^1} ,f_n \in \mathbb{L^1}$. Moreover, $|f_n| \leq K|f|$ a.e on $[a,b]$. If I can prove that $f_n$ converges to $0$ pointwise a.e, I'm done. But, that's what I'm struggling to cope with. I know I have to use some mode of convergence but not sure which one is it.
Is my argument up to this point correct?. Can we argue that $g_n$ converges to $0$ in $\mathbb{L^1}$?.Any help or hint would be appreciated.
This is not true.
Take $g_n(x)=\cos{(2\pi n x)}$ and $f(x)=e^x$ on $[0,1]$ where $c=b=1$