Does $\sin ^n x$ converge uniformly on $[0,\frac{\pi}{2})$? 
Does $f_n(x)=\sin ^n x$ converge uniformly  on $[0,\frac{\pi}{2})$ ?

I know $f_n(x) \rightarrow 0$ pointwise, since $\vert \sin ^n x \vert< 1$. How about the uniform convergence?
Any hint?

Uniform convergence of $(f_n)$ to $f(x) = 0$ would require that
$$
 M_n = \sup \{ |f_n(x) - f(x) | : 0 \le x < \frac \pi 2 \}  
$$
converges to zero. However,  for each $n$
$$
 M_n \ge \lim_{x \to \pi /2} f_n(x) = 1 \, .
$$
