Why is \( c'' \) an Upper Bound for \( U \cap [a, b] \) in Theorem 3.47?

[Math Amateur]

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

 

[GJA]

Hi Peter,

By the definition of $c$, $[p,c]\cap\left(U\cap [a,b]\right)\neq\emptyset$ for all $p<c.$ However, $c''<c$ and, by the choice for $c''$, $$[c'',c]\cap\left(U\cap [a,b]\right)\subset U\cap V\cap [a,b]=\emptyset.$$ Hence, $[c'',c]\cap\left(U\cap [a,b]\right)=\emptyset,$ contradicting the definition of $c$.

 

[Math Amateur]

Hi Peter,

By the definition of $c$, $[p,c]\cap\left(U\cap [a,b]\right)\neq\emptyset$ for all $p<c.$ However, $c''<c$ and, by the choice for $c''$, $$[c'',c]\cap\left(U\cap [a,b]\right)\subset U\cap V\cap [a,b]=\emptyset.$$ Hence, $[c'',c]\cap\left(U\cap [a,b]\right)=\emptyset,$ contradicting the definition of $c$.

Thanks for your reply, GJA ...

It was most helpful ...

Peter