show this sequence of functions is uniformly convergent

**szpengchao** · October 18th 2008, 10:50 AM

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

**ThePerfectHacker** · October 18th 2008, 03:50 PM

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. $f(1)=0$ .

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