Real Analysis - Uniform Convergence

steelphantom

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!

Dick

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