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Real Analysis  Uniform Convergence 
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#1
Apr2908, 07:47 PM

P: 159

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! 


#2
Apr3008, 12:13 AM

Sci Advisor
HW Helper
Thanks
P: 25,250

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



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