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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 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 9153

In the second paragraph of the above proof by Stromberg we read the following:

" ... ... Since \(\displaystyle U\) is open we can choose \(\displaystyle c' \gt c\) such that \(\displaystyle [ c, c' ] \subset U \cap [a, b]\) ... ... "

My question is as follows:

Can someone please demonstrate rigorously why/how ...

\(\displaystyle U\) is open \(\displaystyle \Longrightarrow\) we can choose \(\displaystyle c' \gt c\) such that \(\displaystyle [ c, c' ] \subset U \cap [a, b] \) ... ...

Indeed I can see that ...

\(\displaystyle U\) is open \(\displaystyle \Longrightarrow \exists\) an open ball \(\displaystyle B_r(c) = \ ] c - r, c + r [ \ \subset U\) ... ...

but how do we conclude from here that

\(\displaystyle U\) is open \(\displaystyle \Longrightarrow\) we can choose \(\displaystyle c' \gt c\) such that \(\displaystyle [ c, c' ] \subset U \cap [a, b]\) ... ...

*** EDIT ***

It may be that the solution is to choose \(\displaystyle s \lt r\) so that \(\displaystyle [ c, c + s] \subset U\) where \(\displaystyle c' = c + s\) ... but how do we ensure this interval also belongs to \(\displaystyle [a, b]\) ... ... ?

Help will be appreciated ... ...

Peter

=======================================================================================

Stromberg uses slightly unusual notation for open intervals in \(\displaystyle \mathbb{R}\) and \(\displaystyle \mathbb{R}^{\#} = \mathbb{R} \cup \{ \infty , - \infty \}\) so I am providing access to Stromberg's definition of intervals in \(\displaystyle \mathbb{R}^{ \#} \) ... as follows:

View attachment 9152

Hope that helps ...

Peter

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

I need 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 9153

In the second paragraph of the above proof by Stromberg we read the following:

" ... ... Since \(\displaystyle U\) is open we can choose \(\displaystyle c' \gt c\) such that \(\displaystyle [ c, c' ] \subset U \cap [a, b]\) ... ... "

My question is as follows:

Can someone please demonstrate rigorously why/how ...

\(\displaystyle U\) is open \(\displaystyle \Longrightarrow\) we can choose \(\displaystyle c' \gt c\) such that \(\displaystyle [ c, c' ] \subset U \cap [a, b] \) ... ...

Indeed I can see that ...

\(\displaystyle U\) is open \(\displaystyle \Longrightarrow \exists\) an open ball \(\displaystyle B_r(c) = \ ] c - r, c + r [ \ \subset U\) ... ...

but how do we conclude from here that

\(\displaystyle U\) is open \(\displaystyle \Longrightarrow\) we can choose \(\displaystyle c' \gt c\) such that \(\displaystyle [ c, c' ] \subset U \cap [a, b]\) ... ...

*** EDIT ***

It may be that the solution is to choose \(\displaystyle s \lt r\) so that \(\displaystyle [ c, c + s] \subset U\) where \(\displaystyle c' = c + s\) ... but how do we ensure this interval also belongs to \(\displaystyle [a, b]\) ... ... ?

Help will be appreciated ... ...

Peter

=======================================================================================

Stromberg uses slightly unusual notation for open intervals in \(\displaystyle \mathbb{R}\) and \(\displaystyle \mathbb{R}^{\#} = \mathbb{R} \cup \{ \infty , - \infty \}\) so I am providing access to Stromberg's definition of intervals in \(\displaystyle \mathbb{R}^{ \#} \) ... as follows:

View attachment 9152

Hope that helps ...

Peter

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