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During a preparation for my quals, I encountered the following problem I got stuck on:

Let $\{g_n\}_n$ be a uniformly bounded sequence of functions on $[0, 1]$ which is uniformly Lipschitz, i.e. there is $M>0$ such that $|g_n(x)-g_n(y)|\leq M |x-y|$ for all $n, x, y$. Prove that for any $f \in L^1([0, 1])$ we have $$\lim_{n \rightarrow \infty}\int_0^1 f(x)g_n(x)\sin(2\pi n x) dx=0.$$

I have no idea what to do with this. Intuitively, I can understand why something like $\lim_{n \rightarrow \infty}\int_0^1 g(x)\sin(2\pi n x) dx=0$ for a Lipschitz function $g$ (I can extend this to $L^2$ functions with fancy words such as "orthonormal basis"), but with sequence of functions this seems harder. Also I expect that there will be a trick of some kind, which I cannot figure out.

I left out the first part of the problem which asked to prove that $\lim_{n \rightarrow \infty}\int_a^bg_n(x)\sin(2\pi n x) dx=0$ for any $0 \leq a \leq b \leq 1$. This is clearly a special case (for $f=\chi_{[a, b]}$), but maybe it is useful in the solution of the general one, I don't know.

Thanks in advance for any help.

EDIT: Apparantly, this is a duplicate. I have now found an answer here.

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    If I'm not mistaken, then the first part of the problem it appears as though this limit is true for finite linear combinations of characteristic functions. Then by approximation this limit should be true for continuous functions. Further approximation should give that the limit is true for $L^{1}$ functions. – 2017-01-04

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