Application of the Uniform limit theorem

Calculus Level 2

For nNn\in \mathbb{N}, let fn:[0,π]Rf_n:[0, \pi]\to \mathbb{R} be defined by

fn(x):=sinn(x) f_n(x):= \sin^n (x)

Is the sequence {fn}n=1\{f_n\}_{n=1}^{\infty} uniformly convergent?

Hint: is the limit function continuous?


This problem is often posed in calculus textbooks.
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