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Math Help - Is this statement true or false?

  1. #1
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    Is this statement true or false?

    Suppose that g is a continuous function on the interval [0,1] with g(0)=0. Define a sequence of functions \{f_n\} on the interval [0,1] by f_n(x)=x^ng(x). Does it follow that the sequence of functions \{f_n\} converges uniformly on [0,1]? Either prove this is true or give an example of a function g for which the conclusion fails.

    Any help on this?
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  2. #2
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    Quote Originally Posted by paupsers View Post
    Suppose that g is a continuous function on the interval [0,1] with g(0)=0. Define a sequence of functions \{f_n\} on the interval [0,1] by f_n(x)=x^ng(x). Does it follow that the sequence of functions \{f_n\} converges uniformly on [0,1]? Either prove this is true or give an example of a function g for which the conclusion fails.

    Any help on this?
    The conclusion fails for g(x) := x, for example: because if f_n were to converge uniformly, the limit function would have to be continuous, but it isn't continuous at x=1.
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