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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:
" ... ... $$s + \frac{m}{ 2^n}$$ is an upper bound of $$S$$, for some $$m \in \mathbb{N}$$. Let $$k_n$$ be the smallest such $$m$$ ... ... "Can we argue, based on the above text, that $$s + \frac{m}{ 2^n} = \text{Sup}(S)$$ ... ... ?
Question 2
In the above text we read the following:
" ... ... We then have $$I_n \cap S \ne \emptyset$$ (Why?) ... ... "Is $$I_n \cap S \ne \emptyset$$ because elements such as $$s + \frac{ k_n - x }{ 2^n} , 0 \lt x \lt 1$$ belong to $$I_n \cap S$$ ... for example, the element $$s + \frac{ k_n - 0.5 }{ 2^n} \in I_n \cap S$$?
Is that correct ... if not, then why exactly is $$ I_n \cap S \ne \emptyset$$?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:
" ... ... $$s + \frac{m}{ 2^n}$$ is an upper bound of $$S$$, for some $$m \in \mathbb{N}$$. Let $$k_n$$ be the smallest such $$m$$ ... ... "Can we argue, based on the above text, that $$s + \frac{m}{ 2^n} = \text{Sup}(S)$$ ... ... ?
Question 2
In the above text we read the following:
" ... ... We then have $$I_n \cap S \ne \emptyset$$ (Why?) ... ... "Is $$I_n \cap S \ne \emptyset$$ because elements such as $$s + \frac{ k_n - x }{ 2^n} , 0 \lt x \lt 1$$ belong to $$I_n \cap S$$ ... for example, the element $$s + \frac{ k_n - 0.5 }{ 2^n} \in I_n \cap S$$?
Is that correct ... if not, then why exactly is $$ I_n \cap S \ne \emptyset$$?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