Let $f:[-1,1]\to\mathbb{R}$ be a Riemann integrable (bounded) function and define
$$F_n(x)=\int_{0}^{x}f(\sin(nt))dt,\;\;\;n\in\mathbb{N},x\in\mathbb{R}$$ Prove for any compact set $K$ in $\mathbb{R}$, $F_n$ has a uniformly convergent subsequence on $K$.
$$F_n(x)=\int_{0}^{x}f(\sin(nt))dt,\;\;\;n\in\mathbb{N},x\in\mathbb{R}$$ Prove for any compact set $K$ in $\mathbb{R}$, $F_n$ has a uniformly convergent subsequence on $K$.