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!