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

In summary, Karl R. Stromberg's book "An Introduction to Classical Real Analysis" discusses limits and continuity in real analysis. He uses a slightly unusual notation for open intervals, so I have provided a definition which should help readers following along.
  • #1
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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
 

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  • #2
Peter said:
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
 
  • #3

Hi Peter,

Thank you for reaching out for help with understanding Theorem 3.47 in Stromberg's book. I am also currently studying this chapter and can offer some clarification on the proof.

First, let's define some notation to make things clearer. Let U be an open subset of [a,b], and let c \in U. Since U is open, there exists an open interval B_r(c) = ]c-r, c+r[ \subset U. Now, we can choose s < r such that [c, c+s] \subset B_r(c). This is possible because the interval B_r(c) is open, so we can "shrink" it by choosing a smaller radius s.

Next, we need to show that [c, c+s] \subset [a,b]. Since c \in U \subset [a,b], we know that c \in [a,b]. And since c+s < c+r, we have c+s < b, which means that c+s \in [a,b]. Similarly, since c \in U \subset [a,b], we have c \in [a,b] and since c < c+s, we have c > a, which means that c \in [a,b]. Therefore, [c,c+s] \subset [a,b].

Now, we can choose c' = c+s, and we have shown that c' > c and [c,c'] \subset [a,b]. Also, since [c,c'] \subset B_r(c) \subset U, we have [c,c'] \subset U \cap [a,b], as desired.

I hope this helps clarify the proof for you. Let me know if you have any other questions or need further explanation.

 

Related to Connectedness and Intervals in R .... Stromberg, Theorem 3.47 .... ....

1. What is the definition of connectedness in R?

Connectedness in R refers to the property of a set that cannot be divided into two non-empty disjoint subsets. In other words, a connected set is a set that is continuous and cannot be separated into smaller parts.

2. How is connectedness related to intervals in R?

Intervals in R are a type of connected set. Any interval, whether open, closed, or half-open, is considered a connected set because it cannot be divided into two disjoint subsets. This is due to the fact that intervals are continuous and have no breaks or gaps in them.

3. What is Stromberg's Theorem 3.47 in R?

Stromberg's Theorem 3.47 is a mathematical theorem that states that in a connected set in R, any two points can be connected by a continuous curve. This means that in a connected set, there are no breaks or gaps, and any two points can be reached by a path that does not leave the set.

4. How is Stromberg's Theorem 3.47 useful in mathematics?

Stromberg's Theorem 3.47 is useful in mathematics because it helps to prove the connectedness of a set. By showing that any two points in a set can be connected by a continuous curve, we can conclude that the set is connected. This theorem is also helpful in topology and analysis, where connectedness is an important concept.

5. Can Stromberg's Theorem 3.47 be applied to other mathematical concepts?

Yes, Stromberg's Theorem 3.47 can be applied to other mathematical concepts, such as topological spaces and metric spaces. In these contexts, the theorem is known as the "Path-connectedness Theorem" and states that any two points in a path-connected set can be connected by a continuous path. This is a generalization of the concept of connectedness in R.

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