Let $f_n(x)=\sin^n(x)$. Find the interval of convergence and prove/disprove that $f_n(x)$ converges uniformly in this interval.
My attempt: Finding the pointwise limit, $$\lim_{n\to \infty} \sin^n(x)=\begin{cases} 1 \quad\text{if $x=\pi/2+ 2k\pi$ for $k\in\mathbb{Z}$,}\\ \not\exists \quad\text{if $x=3\pi/2+ 2k\pi$ for $k\in\mathbb{Z}$,}\\ 0 \quad\text{otherwise.}\\ \end{cases}$$ From here I'm not sure how to find the interval of convergence and prove or disprove uniform convergence