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Structure Theorem for the Open Subsets of R... Stromberg, Theorem 3.18 ... ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
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:



Stromberg - 1 - Theorem 3.18 ... ... PART 1 ... .png
Stromberg - 2 - Theorem 3.18 ... ... PART 2 ... .png



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


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Stromberg's notaton for intervals is a bit unusual ... so I am providing the relevant text to explain the notation ... as follows:



Stromberg -  Defn 1.51 ... Intervals of R ... .png



Hope that helps ...

Peter