Application of the Uniform limit theorem

Calculus Level 3

For \(n\in \mathbb{N}\), let \(f_n:[0, \pi]\to \mathbb{R}\) be defined by

\[ f_n(x):= \sin^n (x) \]

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

Hint: is the limit function continuous?


This problem is often posed in calculus textbooks.
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