I want to check whether the following function is uniformly converges: $f_n(x)=n\cos^nx\sin x$ for $x \in \left[0,\frac{\pi}{2} \right]$.
I proved that the $\lim \limits_{n \to \infty}f_{n}(x)=0$ for every $x$. I'd love your help with the uniformly continues convergence. I always get confused with it. I already showed that $|f_n(x) - 0|< \epsilon$. What else should I show or how should I refute the claim?
Thanks a lot.