Real Analysis - Uniform Convergence


by steelphantom
Tags: analysis, convergence, real, uniform
steelphantom
steelphantom is offline
#1
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!
Phys.Org News Partner Science news on Phys.org
SensaBubble: It's a bubble, but not as we know it (w/ video)
The hemihelix: Scientists discover a new shape using rubber bands (w/ video)
Microbes provide insights into evolution of human language
Dick
Dick is offline
#2
Apr30-08, 12:13 AM
Sci Advisor
HW Helper
Thanks
P: 25,175
Of course, it's right. You knew that.


Register to reply

Related Discussions
Uniform Convergence Calculus & Beyond Homework 7
Real Analysis proof help, convergence Calculus & Beyond Homework 11
Real Analysis - Radius of Convergence Calculus & Beyond Homework 9
Real Analysis- least upper bound and convergence General Math 4
Real analysis- Convergence/l.u.b Calculus & Beyond Homework 5