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

Question

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 $\lim_{n\to \infty} f_n (x) = 1$ but if $x \in (0,\pi)$, then $\lim_{n\to \infty} f_n(x) = 0$.

I tried to explain that $\lim_{x\to 0^+}\{ \lim_{n\to\infty} f_n(x)\} = 0$ 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?

Answer 1

A uniformly convergent sequence of continuous functions has a continuous limit.

Notice that the limit of $$ \lim_{n\to \infty} f_n (x) = \begin{cases} 1 \\ 0\\ -1 \end{cases}$$ So by the contrapositive, because the limit is not continuous, the sequence is not uniformly convergent.