I want to prove that $f_{n}(x)=sin(\sqrt{x+4\pi^2 n^2})$ is uniformly convergent in $x\in(0,b)$. I have already seen that it is pointwise convergent with limit $0$. But now I have to look for an upper bound for $|f_{n}(x)|$ for $x\in(0,b)$.
$$sup_{x\in(0,b)}|sin(\sqrt{x+4\pi^2 n^2})|$$
I cannot find an upper bound and I'm starting to suspect that it is not uniformly convergent. I tried deriving but this doesn't give me any information.