Real Analysis - Uniform Convergence

  1. 1. The problem statement, all variables and given/known data
    Prove that if fn -> f uniformly on a set S, and if gn -> g uniformly on S, then fn + gn -> f + g uniformly on S.

    2. Relevant equations

    3. The attempt at a solution
    fn -> f uniformly means that |fn(x) - f(x)| < [tex]\epsilon[/tex]/2 for n > N_1.
    gn -> g uniformly means that |gn(x) - g(x)| < [tex]\epsilon[/tex]/2 for n > N_2.

    By the triangle inequality, we have |fn(x) - f(x) + gn(x) - g(x)| <= |fn(x) - f(x)| + |gn(x) - g(x)| < [tex]\epsilon[/tex]/2 + [tex]\epsilon[/tex]/2 = [tex]\epsilon[/tex].

    This implies |[fn(x) + gn(x)] - [f(x) + g(x)]| < [tex]\epsilon[/tex] for n > N_1, N_2.

    Therefore fn + gn -> f + g uniformly on S.

    Is this correct? I'm pretty confident it's right, but I just want to make sure. Thanks!
     
  2. jcsd
  3. Dick

    Dick 25,887
    Science Advisor
    Homework Helper

    Of course, it's right. You knew that.
     
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