Connectedness and Intervals in R .... Stromberg, 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 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

 

[HallsofIvy]

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:












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 \(\displaystyle [ c, c' ] \subset U \cap [a, b]\) ... …

Take c' to be the smaller of c+ r/2 and (c+ b)/2. Then c< c'< c+ r so is in U and c' is half way between c and b so c' is in [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:









Hope that helps ...

Peter