Let {$g_{n}$}be a bounded sequence of functions on $[0,1]$ which is uniformly Lipschitz. That is, there is a constant $M$ (independent of $n$) such that for all $n$, $|g_{n}(x)-g_n(y)|\leq M|x-y|$ for all $x,y\in [0,1]$ and $|g_{n}(x)|\leq M$ for all $x\in [0,1]$. Then I have the following two questions:
(a) prove for all any $0\leq a\leq b\leq 1$, $\lim_{n\rightarrow \infty } \int_{a}^{b}g_{n}(x)\sin (2n\pi x)\,dx=0. $ (b) prove that for any $f\in L^{1}[0,1]$, $\lim_{n\rightarrow \infty } \int_{0}^{1}f(x)g_{n}(x)\sin (2n\pi x)\,dx=0.$