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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 further 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:
View attachment 9149
View attachment 9150
At about the middle of the above proof by Stromberg we read the following:
" ... ... Otherwise enumerate $$\mathscr{V}$$ as $$\{ V_k \}_{ k = 1 }^{ \infty }$$. ... ... "I am wondering what are the $$V_k$$ ... are they elements of $$\mathscr{V}$$ (... that is, the $$U_B$$) ... or are they sets of some kind ... ... can someone please explain and elucidate the nature of the $$V_k$$ ...
... indeed maybe the $$V_k$$ are just equal to the $$U_B$$ ... in that case why not enumerate $$\mathscr{V}$$ as $$\{ U_{ B_k } \}_{ k = 1 }^{ \infty }$$ ...
Hope someone can help ...
Peter
I am focused on Chapter 3: Limits and Continuity ... ...
I need further 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:
View attachment 9149
View attachment 9150
At about the middle of the above proof by Stromberg we read the following:
" ... ... Otherwise enumerate $$\mathscr{V}$$ as $$\{ V_k \}_{ k = 1 }^{ \infty }$$. ... ... "I am wondering what are the $$V_k$$ ... are they elements of $$\mathscr{V}$$ (... that is, the $$U_B$$) ... or are they sets of some kind ... ... can someone please explain and elucidate the nature of the $$V_k$$ ...
... indeed maybe the $$V_k$$ are just equal to the $$U_B$$ ... in that case why not enumerate $$\mathscr{V}$$ as $$\{ U_{ B_k } \}_{ k = 1 }^{ \infty }$$ ...
Hope someone can help ...
Peter
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