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

Peter

Well-known member
MHB Site Helper
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:

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:

Hope that helps ...

Peter