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Math Help - uniformly convergent

  1. #1
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    uniformly convergent

    Let f_n : \mathbb{R} \rightarrow \mathbb{R} be a sequence of functions that is unifromly convergent in \mathbb{R} to funcion f: \mathbb{R} \rightarrow \mathbb{R}. For n \in \mathbb{N} we define:
    g_n(x)=\exp (-(f_n(x))^2), \ \ \ g(x)=\exp(-(f(x))^2)
    h_n(x)=(f_n(x))^2, \ \ \ h(x)=(f(x))^2.

    Does sequence g_n uniformly convergent in \mathbb{R} to function g?
    Does sequence h_n uniformly convergent in \mathbb{R} to function h?
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  2. #2
    Super Member girdav's Avatar
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    Re: uniformly convergent

    For the first question, write e^{-f_n(x)^2}-e^{-f(x)^2} as \int_{f(x)}^{f_n(x)}-2te^{-t^2}dt and use the fact that the map t\mapsto te^{-t^2} is bounded on the real line.
    For the second question, consider f_n(x)=x+\frac 1n.
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