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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 $$U$$ is open we can choose $$c' \gt c$$ such that $$[ c, c' ] \subset U \cap [a, b]$$ ... ... "
My question is as follows:
Can someone please demonstrate rigorously why/how ...
$$U$$ is open $$\Longrightarrow$$ we can choose $$c' \gt c$$ such that $$[ c, c' ] \subset U \cap [a, b] $$ ... ...
Indeed I can see that ...
$$U$$ is open $$\Longrightarrow \exists$$ an open ball $$B_r(c) = \ ] c - r, c + r [ \ \subset U$$ ... ...but how do we conclude from here that
$$U$$ is open $$\Longrightarrow$$ we can choose $$c' \gt c$$ such that $$[ c, c' ] \subset U \cap [a, b]$$ ... ...*** EDIT ***
It may be that the solution is to choose $$s \lt r$$ so that $$[ c, c + s] \subset U$$ where $$c' = c + s$$ ... but how do we ensure this interval also belongs to $$[a, b]$$ ... ... ?
Help will be appreciated ... ...
Peter
=======================================================================================Stromberg uses slightly unusual notation for open intervals in $$\mathbb{R}$$ and $$\mathbb{R}^{\#} = \mathbb{R} \cup \{ \infty , - \infty \}$$ so I am providing access to Stromberg's definition of intervals in $$\mathbb{R}^{ \#} $$ ... 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 $$U$$ is open we can choose $$c' \gt c$$ such that $$[ c, c' ] \subset U \cap [a, b]$$ ... ... "
My question is as follows:
Can someone please demonstrate rigorously why/how ...
$$U$$ is open $$\Longrightarrow$$ we can choose $$c' \gt c$$ such that $$[ c, c' ] \subset U \cap [a, b] $$ ... ...
Indeed I can see that ...
$$U$$ is open $$\Longrightarrow \exists$$ an open ball $$B_r(c) = \ ] c - r, c + r [ \ \subset U$$ ... ...but how do we conclude from here that
$$U$$ is open $$\Longrightarrow$$ we can choose $$c' \gt c$$ such that $$[ c, c' ] \subset U \cap [a, b]$$ ... ...*** EDIT ***
It may be that the solution is to choose $$s \lt r$$ so that $$[ c, c + s] \subset U$$ where $$c' = c + s$$ ... but how do we ensure this interval also belongs to $$[a, b]$$ ... ... ?
Help will be appreciated ... ...
Peter
=======================================================================================Stromberg uses slightly unusual notation for open intervals in $$\mathbb{R}$$ and $$\mathbb{R}^{\#} = \mathbb{R} \cup \{ \infty , - \infty \}$$ so I am providing access to Stromberg's definition of intervals in $$\mathbb{R}^{ \#} $$ ... as follows:
View attachment 9152
Hope that helps ...
Peter
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