(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that if f_{n}-> f uniformly on a set S, and if g_{n}-> g uniformly on S, then f_{n}+ g_{n}-> f + g uniformly on S.

2. Relevant equations

3. The attempt at a solution

f_{n}-> f uniformly means that |f_{n}(x) - f(x)| < [tex]\epsilon[/tex]/2 for n > N_1.

g_{n}-> g uniformly means that |g_{n}(x) - g(x)| < [tex]\epsilon[/tex]/2 for n > N_2.

By the triangle inequality, we have |f_{n}(x) - f(x) + g_{n}(x) - g(x)| <= |f_{n}(x) - f(x)| + |g_{n}(x) - g(x)| < [tex]\epsilon[/tex]/2 + [tex]\epsilon[/tex]/2 = [tex]\epsilon[/tex].

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

Therefore f_{n}+ g_{n}-> 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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# Real Analysis - Uniform Convergence

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