Is $f_n (x)$ is uniformly convergent on $[0, \pi /2]$?

Question

Is $f_n(x) = \cos^n x(1-\cos^nx)$ is uniformly convergent on $[0, \pi/2]$?

My attempt : I thinks yes. here

Sup $\{ |f_n(x) | : x \in [0, \pi/2]\}= 0$ so $f_n(x) = \cos^n x(1-\cos^nx)$ converge uniformly on $[0,\pi/2]$

Is its true ?

Answer 1

No, it does not converge uniformly. It converges pointwise to the null function, but$$(\forall n\in\mathbb N):f_n\left(\arccos\left(\sqrt[n]{\frac12}\right)\right)=\frac14.$$