I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).(adsbygoogle = window.adsbygoogle || []).push({});

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:

My 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

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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 ...

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# I Supremum Property, Archimedean Property, Nested Intervals

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