Let $ g(x) $ be a continuous periodic function of period 1 on $\mathbb{R}$. Prove that for any integrable function $f(x)$ on $[0,1]$,
$$ \lim_{n \to \infty}\int_0^{1}f(x)g(nx)dx= \int_0^{1}f(x)dx \int_0^{1}g(x)dx.$$
Any help is appreciated.
Let $ g(x) $ be a continuous periodic function of period 1 on $\mathbb{R}$. Prove that for any integrable function $f(x)$ on $[0,1]$,
$$ \lim_{n \to \infty}\int_0^{1}f(x)g(nx)dx= \int_0^{1}f(x)dx \int_0^{1}g(x)dx.$$
Any help is appreciated.