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Real Analysis - Uniform Convergence

by steelphantom
Tags: analysis, convergence, real, uniform
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Apr29-08, 07:47 PM
P: 159
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!
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Of course, it's right. You knew that.

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