Uniform Convergence of $f_n(x)= \cos^nx(1-\cos^nx)$

Question

$$ \text{Is } f_n(x)= \cos^nx(1-\cos^nx) \text{ uniformly convergent on } \left[0,\frac{ \pi}{2}\right] \text{ ? }$$ I solved the question for pointwise convergence and found that $$ f_n(x)\to 0 \space \space \forall \space x \text{ on } \left[0,\frac{\pi}{2} \right] $$ Now i have to show that the series of functions are not uniformly convergent to zero . My question is that what value should I take for $$ x $$ such that it belongs to the $$ \left[0,\frac{ \pi}{2}\right] $$ and such that the function is not uniformly convergent .

Answer 1

Hint: Your $x$ should depend upon $n$ if you want to show non-uniform convergence this way.

You could e.g. use the IVT to show that for any $n$ there is $x$ (depending upon $n$) so that $\cos^n(x)=1/2$.