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- Jun 22, 2012

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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:

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:

Hope that helps ...

Peter