Real Analysis - Uniform Convergence

by steelphantom
Tags: analysis, convergence, real, uniform
 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)| < $$\epsilon$$/2 for n > N_1. gn -> g uniformly means that |gn(x) - g(x)| < $$\epsilon$$/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)| < $$\epsilon$$/2 + $$\epsilon$$/2 = $$\epsilon$$. This implies |[fn(x) + gn(x)] - [f(x) + g(x)]| < $$\epsilon$$ 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!
 HW Helper Sci Advisor Thanks P: 24,423 Of course, it's right. You knew that.

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