Real Analysis - Uniform Convergence

In summary, the conversation discusses proving that if two sequences converge uniformly on a set, their sum will also converge uniformly on that same set. This is shown by using the triangle inequality and the given definitions of uniform convergence. The conclusion is that fn + gn -> f + g uniformly on S. The correctness of the proof is confirmed by the expert.
  • #1
steelphantom
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Homework Statement


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.

Homework Equations



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!
 
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  • #2
Of course, it's right. You knew that.
 

1. What is uniform convergence in real analysis?

Uniform convergence in real analysis is a type of convergence in which a sequence of functions converges to a limit function in such a way that the convergence is independent of the choice of the point at which the functions are evaluated. In other words, the convergence is uniform across the entire domain of the functions.

2. How is uniform convergence different from pointwise convergence?

Pointwise convergence is a type of convergence in which a sequence of functions converges to a limit function at each individual point in the domain. In contrast, uniform convergence requires that the functions converge to the limit function at the same rate across the entire domain.

3. What is the importance of uniform convergence in real analysis?

Uniform convergence is important in real analysis because it allows us to make stronger statements about the convergence of a sequence of functions. It also allows us to interchange limits and integrals, which is crucial for many applications in mathematics and physics.

4. How do we test for uniform convergence of a sequence of functions?

There are several tests for uniform convergence, including the Weierstrass M-test, the Cauchy criterion, and the Dini's theorem. These tests can be used to determine whether a sequence of functions converges uniformly or not.

5. Can a sequence of uniformly convergent functions converge to a non-continuous function?

No, if a sequence of functions is uniformly convergent, then the limit function must also be continuous. This is because uniform convergence guarantees that the functions are converging at the same rate across the entire domain, which implies that the limit function will also be continuous.

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