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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

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At about the middle of the above proof by Stromberg we read the following:

" ... ... Otherwise enumerate \(\displaystyle \mathscr{V}\) as \(\displaystyle \{ V_k \}_{ k = 1 }^{ \infty }\). ... ... "I am wondering what are the \(\displaystyle V_k\) ... are they elements of \(\displaystyle \mathscr{V}\) (... that is, the \(\displaystyle U_B\)) ... or are they sets of some kind ... ...

... indeed maybe the \(\displaystyle V_k\) are just equal to the \(\displaystyle U_B\) ... in that case why not enumerate \(\displaystyle \mathscr{V}\) as \(\displaystyle \{ U_{ B_k } \}_{ k = 1 }^{ \infty }\) ...

Hope someone can help ...

Peter

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

I need

*help in order to fully understand the proof of Theorem 3.43 on pages 105-106 ... ... Theorem 3.43 and its proof read as follows:***further**View attachment 9149

View attachment 9150

At about the middle of the above proof by Stromberg we read the following:

" ... ... Otherwise enumerate \(\displaystyle \mathscr{V}\) as \(\displaystyle \{ V_k \}_{ k = 1 }^{ \infty }\). ... ... "I am wondering what are the \(\displaystyle V_k\) ... are they elements of \(\displaystyle \mathscr{V}\) (... that is, the \(\displaystyle U_B\)) ... or are they sets of some kind ... ...

*\(\displaystyle V_k\) ...***can someone please explain and elucidate the nature of the**... indeed maybe the \(\displaystyle V_k\) are just equal to the \(\displaystyle U_B\) ... in that case why not enumerate \(\displaystyle \mathscr{V}\) as \(\displaystyle \{ U_{ B_k } \}_{ k = 1 }^{ \infty }\) ...

Hope someone can help ...

Peter

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