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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 $$U \cap V \cap [a, b] \ \subset \ U \cap V \cap S = \emptyset$$, and so $$c''$$ is an upper bound for $$U \cap [a, b]$$ ... ... " My question is as follows:
Can someone please demonstrate rigorously how/why ...
$$U \cap V \cap [a, b] \ \subset \ U \cap V \cap S = \emptyset \Longrightarrow c''$$ is an upper bound for $$U \cap [a, b]$$ ...
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 $$U \cap V \cap [a, b] \ \subset \ U \cap V \cap S = \emptyset$$, and so $$c''$$ is an upper bound for $$U \cap [a, b]$$ ... ... " My question is as follows:
Can someone please demonstrate rigorously how/why ...
$$U \cap V \cap [a, b] \ \subset \ U \cap V \cap S = \emptyset \Longrightarrow c''$$ is an upper bound for $$U \cap [a, b]$$ ...
Help will be appreciated ...
Peter
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