Understanding Stromberg's Theorem 3.18 & its Proof

[Math Amateur]

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:
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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:
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Hope that helps ...

Peter

 

[jvicens]

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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!
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