Another Convergence Question

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In summary, we are given a uniformly bounded sequence of Riemann-integrable functions on [a,b], and we need to prove the existence of a uniformly convergent subsequence of the sequence of integrals of these functions. To do this, we use the Arzela-Ascoli Theorem, which states that a sequence of uniformly bounded and equicontinuous functions on a compact set must have a uniformly convergent subsequence. We have shown that the sequence is uniformly bounded, and now we just need to show that it is equicontinuous. This can be done using the fact that the functions are Riemann-integrable and the Mean Value Theorem.
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e(ho0n3
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Homework Statement
Let [tex]\{f_n\}[/tex] be a uniformly bounded sequence of functions which are Riemann-integrable on [a,b]. Let
[tex]F_n(x) = \int_a^x f_n(t) \, dt[/tex]
Prove that there exists a subsequence of [tex]\{F_n\}[/tex] which converges uniformly on [a,b].

The attempt at a solution
I was thinking, since [tex]\{f_n\}[/tex] is uniformly bounded, there is an M such that [tex]F_n(x) \le M(x - a) \le M(b - a)[/tex] for all n, for all x. Now this automatically means that [tex]\{F_n\}[/tex] converges uniformly right? But then if [tex]\{F_n\}[/tex] converges uniformly, why is the problem requesting for a subsequence? I must have done something wrong.
 
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  • #2
The f_n aren't given to be convergent. Sure the F_n are bounded, but that certainly doesn't mean they are convergent. Can't you think of a theorem that guarantees the existence of a uniformly convergent subsequence in a family of functions? What are it's premises?
 
  • #3
You mean the Arzela-Ascoli Theorem right? The Wikipedia article actually states what I'm trying to prove I think:

For example, the theorem's hypotheses are satisfied by a uniformly bounded sequence of differentiable functions with uniformly bounded derivatives.

the hypotheses being that the sequence must be uniformly bounded and equicontinuous. I have already shown that the sequence is uniformly bounded right? So all I need to show is that it is equicontinuous. I think I can handle that.
 
  • #4
e(ho0n3 said:
You mean the Arzela-Ascoli Theorem right? The Wikipedia article actually states what I'm trying to prove I think:



the hypotheses being that the sequence must be uniformly bounded and equicontinuous. I have already shown that the sequence is uniformly bounded right? So all I need to show is that it is equicontinuous. I think I can handle that.

You've got it.
 

1. What is "Another Convergence Question"?

"Another Convergence Question" is a term used to describe a mathematical or scientific problem that involves finding the point at which two or more lines, functions, or systems meet or intersect.

2. What is the importance of studying convergence?

Convergence is a fundamental concept in mathematics and science, as it allows us to determine the behavior and limits of various systems. It is essential for understanding and predicting phenomena in fields such as calculus, physics, and economics.

3. What are some common examples of convergence?

Some common examples of convergence include the convergence of a series, the convergence of a limit, and the convergence of an algorithm. Other examples include the convergence of genetic traits in a population, the convergence of evolutionary processes, and the convergence of technological advancements.

4. What are the different types of convergence?

There are several types of convergence, including pointwise convergence, uniform convergence, and absolute convergence. Pointwise convergence refers to the convergence of a sequence of values at a specific point, while uniform convergence refers to the convergence of a sequence of values across an entire interval. Absolute convergence is a stricter form of convergence that requires the series to converge regardless of the order in which the terms are added.

5. How is convergence used in real-world applications?

Convergence has many practical applications, such as in data analysis and optimization problems. It is also used in engineering and technology to improve the performance and efficiency of systems. In addition, convergence is essential in the study of complex systems, such as weather patterns, stock market trends, and population dynamics.

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