Understanding Stromberg's Theorem 3.18 & its Proof

In summary, Stromberg's Theorem 3.18 is a mathematical theorem that states that the limit function of a sequence of continuous functions on a compact metric space is also continuous if the sequence converges uniformly. The proof involves using the definition of uniform convergence and the fact that a uniformly convergent sequence of continuous functions on a compact metric space must have a continuous limit function. This theorem is commonly used in real analysis, functional analysis, and other fields where continuous functions and their limits are important. Examples of its applications include proving the Weierstrass approximation theorem and the Arzelà-Ascoli theorem. However, this theorem has limitations as it only applies to sequences on compact metric spaces and does not guarantee the existence of a limit function
  • #1
Math Amateur
Gold Member
MHB
3,998
48
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...

I am focused on Chapter 3: Limits and Continuity ... ...

I need help in order to fully understand the proof of Theorem 3.18 on pages 98-99 ... ... Theorem 3.18 and its proof read as follows:
View attachment 9119
View attachment 9120
In the above proof by Stromberg we read the following:

" ... ... If \(\displaystyle a_x \lt t \leq x\), then \(\displaystyle t\) is not a lower bound for \(\displaystyle A_x\), and so there exists an \(\displaystyle \alpha \in A_x\) such that \(\displaystyle t \in \ ] \alpha, x ] \subset V\); whence \(\displaystyle ] a_x, x ] \subset V\) ... ... "
Can someone please explain (demonstrate rigorously...) how the existence of an \(\displaystyle \alpha \in A_x\) such that \(\displaystyle t \in \ ] \alpha, x ] \subset V\) implies \(\displaystyle ] a_x, x ] \subset V\) ... ...Help will be appreciated ...

Peter======================================================================================Stromberg's notaton for intervals is a bit unusual ... so I am providing the relevant text to explain the notation ... as follows:
View attachment 9121
Hope that helps ...

Peter
 

Attachments

  • Stromberg - 1 - Theorem 3.18 ... ... PART 1 ... .png
    Stromberg - 1 - Theorem 3.18 ... ... PART 1 ... .png
    6.3 KB · Views: 90
  • Stromberg - 2 - Theorem 3.18 ... ... PART 2 ... .png
    Stromberg - 2 - Theorem 3.18 ... ... PART 2 ... .png
    42.8 KB · Views: 91
  • Stromberg -  Defn 1.51 ... Intervals of R ... .png
    Stromberg - Defn 1.51 ... Intervals of R ... .png
    10.9 KB · Views: 82
Physics news on Phys.org
  • #2


======================================================================================

Dear Peter,

Thank you for reaching out for help with Theorem 3.18 in Stromberg's book. This theorem is an important concept in classical real analysis and understanding its proof will greatly enhance your understanding of limits and continuity.

To begin, let's define some notation in Stromberg's proof. The notation ]a, b[ represents the open interval from a to b, meaning that the interval includes all real numbers between a and b, but not including a and b themselves. Similarly, ]a, b] represents the half-open interval from a to b, including a but not b, and [a, b] represents the closed interval from a to b, including both a and b.

Now, let's break down the proof step by step. We are given that a_x < t ≤ x, meaning that t is between a_x and x. This also means that t is not a lower bound for A_x, as if it were, then a_x would be the greatest lower bound for A_x and t would not be between a_x and x. Therefore, there exists an α ∈ A_x such that t ∈ ]α, x], meaning that t is between α and x. This also implies that ]a_x, x] is a subset of ]α, x], as every number between a_x and x is also between α and x.

Since t is between a_x and x, and α is between a_x and x, we can conclude that t is also between α and x. In other words, ]a_x, x] is a subset of ]α, x]. And since t is also in the interval ]α, x], we can say that ]a_x, x] is a subset of V, as V is defined as the set of all numbers between α and x.

Therefore, we have shown that ]a_x, x] is a subset of V, which is what we wanted to prove. This also means that for any number t between a_x and x, there exists an α ∈ A_x such that t ∈ ]α, x] and ]a_x, x] is a subset of V. This is the key idea behind Theorem 3.18.

I hope this explanation helps you better understand the proof. Please let me know if you have any further questions or need clarification on any of the steps. Keep up the good work in your studies!
[
 

FAQ: Understanding Stromberg's Theorem 3.18 & its Proof

What is Stromberg's Theorem 3.18?

Stromberg's Theorem 3.18 is a mathematical theorem that states that if a function is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there exists at least one point c within the interval (a,b) where the derivative of the function is equal to the average rate of change of the function over the interval [a,b].

Why is Stromberg's Theorem 3.18 important?

Stromberg's Theorem 3.18 is important because it provides a powerful tool for analyzing the behavior of continuous and differentiable functions. It allows us to determine the existence of a point where the derivative of the function is equal to the average rate of change, which can provide valuable insights into the behavior of the function.

What is the proof of Stromberg's Theorem 3.18?

The proof of Stromberg's Theorem 3.18 involves using the Mean Value Theorem and the Intermediate Value Theorem to show that there exists at least one point c in the interval (a,b) where the derivative of the function is equal to the average rate of change. This is done by considering the function f(x) = g(x) - h(x), where g(x) is the average rate of change of the function and h(x) is the derivative of the function.

How is Stromberg's Theorem 3.18 used in real-life applications?

Stromberg's Theorem 3.18 has various real-life applications, such as in economics, physics, and engineering. It can be used to analyze the behavior of functions that represent real-world phenomena, such as the rate of change of a stock price over time or the velocity of a moving object. It also has applications in optimization problems, where it can be used to find the optimal value of a function.

Are there any limitations to Stromberg's Theorem 3.18?

One limitation of Stromberg's Theorem 3.18 is that it only applies to continuous and differentiable functions. It cannot be used for functions that are not differentiable, such as step functions. Additionally, the theorem only guarantees the existence of at least one point where the derivative is equal to the average rate of change, but it does not provide any information about the location or uniqueness of this point.

Back
Top