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I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with Theorem 2.1.45 concerning the Supremum Property (AoC), the Archimedean Property, and the Nested Intervals Theorem ... ...
Theorem 2.1.45 reads as follows:View attachment 7166
https://www.physicsforums.com/attachments/7167My questions regarding the above text from Sohrab are as follows:Question 1
In the above text we read the following:
" ... ... \(\displaystyle s + \frac{m}{ 2^n}\) is an upper bound of \(\displaystyle S\), for some \(\displaystyle m \in \mathbb{N}\). Let \(\displaystyle k_n\) be the smallest such \(\displaystyle m\) ... ... "Can we argue, based on the above text, that \(\displaystyle s + \frac{m}{ 2^n} = \text{Sup}(S)\) ... ... ?
Question 2
In the above text we read the following:
" ... ... We then have \(\displaystyle I_n \cap S \ne \emptyset\) (Why?) ... ... "Is \(\displaystyle I_n \cap S \ne \emptyset\) because elements such as \(\displaystyle s + \frac{ k_n - x }{ 2^n} , 0 \lt x \lt 1\) belong to \(\displaystyle I_n \cap S\) ... for example, the element \(\displaystyle s + \frac{ k_n - 0.5 }{ 2^n} \in I_n \cap S\)?
Is that correct ... if not, then why exactly is \(\displaystyle I_n \cap S \ne \emptyset\)?Hope someone can help ...
Peter==========================================================================================The above theorem concerns the Supremum Property, the Archimedean Property and the Nested Intervals Theorem ... so to give readers the context and notation regarding the above post I am posting the basic information on these properties/theorems ...https://www.physicsforums.com/attachments/7168
https://www.physicsforums.com/attachments/7169
View attachment 7170
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with Theorem 2.1.45 concerning the Supremum Property (AoC), the Archimedean Property, and the Nested Intervals Theorem ... ...
Theorem 2.1.45 reads as follows:View attachment 7166
https://www.physicsforums.com/attachments/7167My questions regarding the above text from Sohrab are as follows:Question 1
In the above text we read the following:
" ... ... \(\displaystyle s + \frac{m}{ 2^n}\) is an upper bound of \(\displaystyle S\), for some \(\displaystyle m \in \mathbb{N}\). Let \(\displaystyle k_n\) be the smallest such \(\displaystyle m\) ... ... "Can we argue, based on the above text, that \(\displaystyle s + \frac{m}{ 2^n} = \text{Sup}(S)\) ... ... ?
Question 2
In the above text we read the following:
" ... ... We then have \(\displaystyle I_n \cap S \ne \emptyset\) (Why?) ... ... "Is \(\displaystyle I_n \cap S \ne \emptyset\) because elements such as \(\displaystyle s + \frac{ k_n - x }{ 2^n} , 0 \lt x \lt 1\) belong to \(\displaystyle I_n \cap S\) ... for example, the element \(\displaystyle s + \frac{ k_n - 0.5 }{ 2^n} \in I_n \cap S\)?
Is that correct ... if not, then why exactly is \(\displaystyle I_n \cap S \ne \emptyset\)?Hope someone can help ...
Peter==========================================================================================The above theorem concerns the Supremum Property, the Archimedean Property and the Nested Intervals Theorem ... so to give readers the context and notation regarding the above post I am posting the basic information on these properties/theorems ...https://www.physicsforums.com/attachments/7168
https://www.physicsforums.com/attachments/7169
View attachment 7170