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Math Help - Pointwise and uniform convergence proof?

  1. #1
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    Pointwise and uniform convergence proof?

    we have two sequences of functions on an interval I in the real line.
    individually, these functions converge uniformly on I.

    how do i prove that the product of these functions converges pointwise on I?


    -----------
    i know the definitions of each form of convergence.
    also know that if we let fn be a series of continuous functions that uniformly converges to a function f. Then f is continuous.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Pointwise and uniform convergence proof?

    Quote Originally Posted by cassius View Post
    we have two sequences of functions on an interval I in the real line. individually, these functions converge uniformly on I. how do i prove that the product of these functions converges pointwise on I?
    The uniform convergence is irrelevant. Take into account that

    \lim_{n\to \infty}f_n(x)=f(x)\;\wedge \;\lim_{n\to \infty}g_n(x)=g(x)\quad \forall x\in I\Rightarrow

    \lim_{n\to \infty}f_n(x)g_n(x)=\lim_{n\to \infty}f_n(x)\cdot \lim_{n\to \infty}g_n(x)=f(x)g(x)\quad \forall x\in I
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