# show this sequence of functions is uniformly convergent

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• October 18th 2008, 10:50 AM
szpengchao
show this sequence of functions is uniformly convergent
Suppose $f : [0,1]\rightarrow R$ is continuous.
Show that the sequence $(x^{n}f(x))$ is uniformly convergent on [0,1] iff f(1)=0
• October 18th 2008, 03:50 PM
ThePerfectHacker
Quote:

Originally Posted by szpengchao
Suppose $f : [0,1]\rightarrow R$ is continuous.
Show that the sequence $(x^{n}f(x))$ is uniformly convergent on [0,1] iff f(1)=0

There is $M$ so that $|f|\leq M$. This means $|x^n f(x)| \leq Mx^n \to 0$, uniformly, for $x\in [0,1)$. Thus, to have uniform convergence we require that $(1)^n f(1) \to 0$ to i.e. $f(1)=0$.