TITLE: Uniform convergence of $x^n$ on $(-1,1)$.
QUESTION [3 upvotes]: Let $f_n:(-1,1) \rightarrow \mathbb R: x\mapsto x^n$. I want to show that the sequence $(f_n)_{n=0}^\infty$ can not converge uniformly. How can i prove that ?

REPLY [2 votes]: Note that $\lim_{n\to\infty}(1-\frac{1}{n})^n=\frac{1}{e}$,which is a contradiction of uniform convergence,since the pointwise limit is zero
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