# Supremum Property (AoC) .... etc .... Another question/Issue

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• Math Amateur
In summary, the conversation is about a specific theorem in a real analysis book and the confusion surrounding the logic and notation used in the proof. The conversation delves into the definitions and properties of the Supremum Property, the Archimedean Property, and the Nested Intervals Theorem to explain the steps in the proof. The conversation ends with the clarification of the logic and notation used in the proof.
Math Amateur
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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 another issue/problem with Theorem 2.1.45 concerning the Supremum Property (AoC), the Archimedean Property, and the Nested Intervals Theorem ... ...

In the above proof by Sohrab, we read the following:

" ... ... Thus while (by definition) ##s + \frac{k_n}{2^n} = s + \frac{2 k_n}{ 2^{n + 1} }## is an upper bound of ##S, s + \frac{ 2 k_n - 2 }{ 2^{n + 1} } = s + \frac{k_n - 1 }{2^n}## is not. Therefore, either ##k_{ n+1 } = 2k_n## or ##k_{ n+1 } = 2k - 1## and ##I_{ n + 1 } \subset I_n## follow. ... ... "I am uncertain and somewhat confused by the logic of the above ...

I am not sure of the argument that either ##k_{ n+1 } = 2k_n## or ##k_{ n+1 } = 2k - 1## ... can someone please explain in simple terms why it is valid ... ..

I am also not sure exactly why ##I_{ n + 1 } \subset I_n## ... if anything it seems to me that ##I_{ n + 1 } = I_n## ... again, can someone please explain in simple terms why ##I_{ n + 1 } \subset I_n## ... that is that ##I_{ n + 1 }## is a proper subset of ##I_n## ... ...

[ ***EDIT*** Just checked and found that Sohrab is using \subset in the sense which includes equality ... so using ##\subset## as meaning ##\subseteq## ... ]Help will be appreciated ...

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 ...

#### Attachments

• Sohrab - 1 - Theorem 2.1.45 ... - PART 1 ... ....png
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• Sohrab - 2 - Theorem 2.1.45 ... - PART 2 ... ....png
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• Sohrab - Axiom of Completeness ... Supremum Property ....png
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• Sohrab - Theorem 2.1.31 - Archimedean Property ... ....png
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• Sohrab - Theorem 2.1.43 ... Nested Intervals Theorem ....png
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Last edited:
Math Amateur said:
In the above proof by Sohrab, we read the following:

" ... ... Thus while (by definition) ##s + \frac{k_n}{2^n} = s + \frac{2 k_n}{ 2^{n + 1} }## is an upper bound of ##S, s + \frac{ 2 k_n - 2 }{ 2^{n + 1} } = s + \frac{k_n - 1 }{2^n}## is not. Therefore, either ##k_{ n+1 } = 2k_n## or ##k_{ n+1 } = 2k - 1## and ##I_{ n + 1 } \subset I_n## follow. ... ... "I am uncertain and somewhat confused by the logic of the above ...

I am not sure of the argument that either ##k_{ n+1 } = 2k_n## or ##k_{ n+1 } = 2k - 1## ... can someone please explain in simple terms why it is valid ... ..

I am also not sure exactly why ##I_{ n + 1 } \subset I_n## ... if anything it seems to me that ##I_{ n + 1 } = I_n## ... again, can someone please explain in simple terms why ##I_{ n + 1 } \subset I_n## ... that is that ##I_{ n + 1 }## is a proper subset of ##I_n##
I assume you accept the statement that ##s + \frac{ 2 k_n - 2 }{ 2^{n + 1} }## is not an upper bound of ##S##. If not, this post won't help.

##k_{n+1}## is defined as the smallest ##m\in\mathbb N## such that ##s+k_{n+1}/2^{n+1}## is an UB for ##S##. We know that ##s+(2k_n-2)/2^{n+1}## is not an UB, therefore it must be too small to be an UB. So we must have ##k_{n+1}> 2k_n-2##.

How much bigger does ##k_{n+1}## have to be? We know that ##2k_n## is big enough. So ##k_{n+1}\leq 2k_n##.

The only integers ##m## that satisfy ##2k_n-2 < m \leq 2k_n## are those two the text names.

Math Amateur
Thanks Andrew ... now understand ...

Peter

## What is the Supremum Property (AoC)?

The Supremum Property, also known as the Axiom of Completeness (AoC), is a fundamental concept in real analysis. It states that every non-empty set of real numbers that is bounded above has a least upper bound, or supremum, which is also a real number. This property is a key component in proving the existence of limits, continuity, and the convergence of sequences and series.

## Why is the Supremum Property important?

The Supremum Property is important because it allows us to work with real numbers in a rigorous and precise manner. It provides a foundation for many concepts in real analysis and allows us to prove the existence of important mathematical objects, such as limits and derivatives. Without this property, many important theorems and results in mathematics would not hold true.

## What is the difference between supremum and maximum?

While both supremum and maximum refer to the largest element in a set, there is a key difference between the two. The supremum of a set is the least upper bound, meaning it is the smallest number that is greater than or equal to all elements in the set. On the other hand, the maximum is the largest element in the set. A set may not have a maximum, but it will always have a supremum if the Supremum Property holds.

## How is the Supremum Property used in mathematical proofs?

The Supremum Property is often used in proofs to show the existence of a certain element or to prove the convergence of a sequence or series. It is also used to define and prove important concepts in mathematics such as continuity and differentiability. In many cases, the Supremum Property is a crucial step in proving mathematical theorems and results.

## Are there any applications of the Supremum Property outside of mathematics?

Yes, the Supremum Property has applications in various fields such as economics, physics, and computer science. In economics, it is used to model consumer preferences and production possibilities. In physics, it is used in the analysis of wave functions and quantum mechanics. In computer science, it is used in algorithms for finding the optimal solution to a problem. The Supremum Property is a powerful tool that has implications in many different areas of study.

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