real analysis – Prove that $f_n(x) = cos^{n} x$ is not uniformly convergent on $[0, pi]$

Prove that $f_n(x) = cos^{n} x$ is not uniformly convergent on $(0, pi)$

Intuitively I can see that if $x=0$ or $x = pi$ then $lim_{nto infty} f_n (x) = 1$ but if $x in (0,pi)$, then $lim_{nto infty} f_n(x) = 0$.

I tried to explain that $lim_{xto 0^+}{ lim_{ntoinfty} f_n(x)} = 0$ but $f_n(0) = 1$ so the limit does not exist which would imply that there is also no possible uniform convergence.

What is the right approach?