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Math Help - Uniform convergence

  1. #1
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    Uniform convergence

    I am asked to show that f_n (x) = x^n is uniformly convergent in (0, 1) but not in [0, 1].

    I proceed by computing the pointwise convergence to f(x) = 0 for x \in [0, 1) and f(x) = 1 for x = 1.

    Then I compute for x \in [0, 1):

    \vert f_n(x) - f(x) \vert < \epsilon \Leftrightarrow \vert x^n \vert < \epsilon \Leftrightarrow x < (\epsilon)^{1/n}

    This is not fullfilled for all x \in (0, 1) (since I can always find an x : (\epsilon)^{1/n} < x < 1) and therefore I have shown that f_n (x) = x^n is not uniformly convergent in (0, 1).

    I am confused since I was asked to show that it is unformly convergent in (0, 1). What is wrong?
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  2. #2
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    Quote Originally Posted by tholan View Post
    I am asked to show that f_n (x) = x^n is uniformly convergent in (0, 1) but not in [0, 1].

    I proceed by computing the pointwise convergence to f(x) = 0 for x \in [0, 1) and f(x) = 1 for x = 1.

    Then I compute for x \in [0, 1):

    \vert f_n(x) - f(x) \vert < \epsilon \Leftrightarrow \vert x^n \vert < \epsilon \Leftrightarrow x < (\epsilon)^{1/n}

    This is not fullfilled for all x \in (0, 1) (since I can always find an x : (\epsilon)^{1/n} < x < 1) and therefore I have shown that f_n (x) = x^n is not uniformly convergent in (0, 1).

    I am confused since I was asked to show that it is unformly convergent in (0, 1). What is wrong?
    The question is wrong. The sequence of functions f_n (x) = x^n is not uniformly convergent in (0, 1), as the above argument correctly shows.
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  3. #3
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    Thank you.
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