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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 3.47 on page 107 ... ... Theorem 3.47 and its proof read as follows:

View attachment 9155In the third paragraph of the above proof by Stromberg we read the following:

" ... ... But \(\displaystyle U \cap V \cap [a, b] \ \subset \ U \cap V \cap S = \emptyset\), and so \(\displaystyle c''\) is an upper bound for \(\displaystyle U \cap [a, b]\) ... ... " My question is as follows:

Can someone please demonstrate rigorously how/why ...

\(\displaystyle U \cap V \cap [a, b] \ \subset \ U \cap V \cap S = \emptyset \Longrightarrow c''\) is an upper bound for \(\displaystyle U \cap [a, b]\) ...

Help will be appreciated ...

Peter

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

I need further help in order to fully understand the proof of Theorem 3.47 on page 107 ... ... Theorem 3.47 and its proof read as follows:

View attachment 9155In the third paragraph of the above proof by Stromberg we read the following:

" ... ... But \(\displaystyle U \cap V \cap [a, b] \ \subset \ U \cap V \cap S = \emptyset\), and so \(\displaystyle c''\) is an upper bound for \(\displaystyle U \cap [a, b]\) ... ... " My question is as follows:

Can someone please demonstrate rigorously how/why ...

\(\displaystyle U \cap V \cap [a, b] \ \subset \ U \cap V \cap S = \emptyset \Longrightarrow c''\) is an upper bound for \(\displaystyle U \cap [a, b]\) ...

Help will be appreciated ...

Peter

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