4:16 AM
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 $K$.
This should be easy, but I could not recall the related theorems
Since $K$ is compact, $F_n(K)$ is compact.

2 hours later…
6:26 AM
0

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...

7:05 AM
$F_n(K)$ is compact, then $\{F_n\}$ is uniformly bounded, since $F_n$ is lipschitz, then the sequence is equicontinuous; then apply A-Z theorem

7:16 AM
A-Z theorem means Arzelàâ€“Ascoli theorem? Where does the shortcut "A-Z" come from?

10 hours later…
5:00 PM
Arzelàâ€“Ascoli theorem, that is what i mean, for some reason, I was saying arzela-zscoli in my head