# Connectedness and Intervals in R .... Stromberg, Theorem 3.47 .... ....

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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

#### Attachments

Last edited:

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