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Math Help - show this sequence of functions is uniformly convergent

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by szpengchao View Post
    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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