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 $K$.
I just took my real analysis qualifying exam yesterday and this problem showed up:
Prove $F_n(x) = \int_0^x f(\sin(nt))dt$ where $f$ is Riemann integrable on $[-1,1]$ converges uniformly on compact subsets of $[0,\infty)$ and find its limit.
There were two previous parts to the problem, showing t...
