How would you show that $(1-x) x^n$ is uniformly convergent on $[0,1]$?
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Is the sum of uniformly convergent functions uniformly convergent? – abiessu Apr 24 '14 at 18:18
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@abiessu Are you mixing up uniform convergence and uniform continuity? Note that $x^n$ and $-x^{n+1}$ are not uniformly convergent on $[0,1]$. – Andrés E. Caicedo Apr 25 '14 at 00:41
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To prove uniform convergence, first, note that $\lim_{n\to \infty} (1-x)x^n=0 $. Now, you need to prove
$$ \sup_{x\in[0,1]}|(1-x)x^n-0|< \epsilon. $$
To find the $\sup$, you can use the derivative test to get that the sup is achieved at point $x=\frac{n}{n+1}$, that gives
$$ \sup_{x\in[0,1]}|(1-x)x^n-0|= \frac{(\frac{n}{n+1})^n}{n+1}<\dots. $$
I leave it here for you to finish the problem. See related technique.
Mhenni Benghorbal
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