Supremum Property (AoC) .... etc .... Yet a further question

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• Math Amateur
In summary, the conversation is about discussing Theorem 2.1.45 from Houshang H. Sohrab's book "Basic Real Analysis" (Second Edition) and its implications for the Supremum Property, the Archimedean Property, and the Nested Intervals Theorem. The conversation includes a question about Sohrab's process and assumptions in the proof of the theorem, specifically regarding the use of examples and justification for certain statements. The expert summarizes the conversation and confirms that the reasoning is valid, providing additional explanation and clarification.
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 yet a further issue/problem 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:

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

" ... ...The Nested Intervals Theorem now implies that ##\bigcap_{ n = 1}^{ \infty } I_n = \{ u \}## for a unique ##u \in \mathbb{R}##. Indeed, if ##u \lt v## and ##u,v \in \bigcap_{ n = 1}^{ \infty } I_n##, then ##v - u \gt \frac{1}{2n}## for some ##n \in \mathbb{N}##, which contradicts ##u, v \in I_n##, since ##I_n## has length ##2^{ -n }##. ... ... "I am unsure of Sohrab's process and assumptions as he is moving through the proof in the above quote ... could someone confirm (or otherwise) my interpretations as follows ... there are essentially 4 questions ( Q1, Q2, Q3 and Q4 respectively ...) ... ...First issue ... ... I assume that when Sohrab writes: "Indeed, if ##u \lt v## ... etc etc ... " ... he is verifying his statement that ##\bigcap_{ n = 1}^{ \infty } I_n = \{ u \}## for a unique ##u \in \mathbb{R}##? Is that right? (Q1) Second issue ... ... when Sohrab writes: "Indeed, if ##u \lt v## ... etc etc ... " ... ... he could have said ##u \gt v## ... but he is just taking ##u \lt v## as an example ... and we are left to infer that ##u \gt v## works similarly ... in other words there is no reason that ##u## is taken as less than ##v## as against taking ##v \lt u## ... ... Is that right? (Q2)

Third issue ... ... Sohrab then asserts that ##v - u \gt \frac{1}{ 2^n }## ... ... and I am assuming this follows because ...

##u \lt v##

##\Longrightarrow v - u \gt 0##

##\Longrightarrow v - u \gt \frac{1}{n}## for some ##n \in \mathbb{N}## ... (Corollary 2.1.32 (b) Archimedean Property ... see scanned text insert below)

##\Longrightarrow v - u \gt \frac{1}{ 2^n }## ... ... ... ( Is this valid? (Q3) ... looks OK ... but justification ?

So indeed ... given we are doing analysis ... how do we justify ##\frac{1}{n} \gt \frac{1}{ 2^n }## or ##2^n \gt n##?

and further ... is my interpretation above for the third issue correct (Q4)

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 ...

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• 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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I'm only going to address Q1 and Q2 in this post. Really, since we know the diameter of the nested sequences converges to 0, we also know such a unique ##u## exists. Probably for pedagogical purposes, he shows that there cannot be two elements in the infinite intersection. Because these two elements ##u, v## are arbitrary anyway, there is no need to show both cases ##u<v## AND ##v<u##.

Math Amateur
With regard to the fact that ##v-u>2^{-n}##, your reasoning is sound. It is easy to show that ##2^n>n## for all natural numbers. First, ##2^0=1>0##. Second, ##2^n>n \implies 2^{n+1}>n+1##. Adding 1 to the antecedent, ##2^n+1>n+1##. Since ##2 \cdot 2^n > 2^n+1## is true whenever ##n>0## (verify this by subtracting ##2^n## and taking ##\log_2##), we know that ##2^{n+1}>n+1## as desired.

This reasoning is a bit tedious, though. It's pretty obvious that ##(1/2^n)_{n \in \mathbb{N}} \to 0##,and any such sequence must get smaller than any given positive number—this is the spirit of the Archimedean property: there are no arbitrarily small numbers. Therefore, I imagine he took for granted that there existed a ##v-u>1/2^n## for some n.

Let me know if you have follow up questions.

Math Amateur
Thanks for your guidance and support Someone2841 ... really helpful ...

Peter

What is the Supremum Property?

The Supremum Property, also known as the Axiom of Completeness, is a fundamental property of real numbers. It states that every non-empty set of real numbers that is bounded above has a least upper bound, also known as the supremum.

How does the Supremum Property relate to the Axiom of Choice?

The Supremum Property is equivalent to the Axiom of Choice, meaning that if one is assumed to be true, the other can be proven to be true. The Supremum Property is often used as an alternative to the Axiom of Choice in mathematical proofs.

Why is the Supremum Property important in mathematics?

The Supremum Property is important because it allows for the existence of limits, continuity, and convergence in real analysis. It also plays a crucial role in the development of calculus, as it provides a way to define and evaluate infinite series.

How is the Supremum Property used in everyday life?

The Supremum Property is used in many real-world applications, such as in economics, finance, and engineering. For example, it is used to find the optimal solution to problems involving maximizing or minimizing a certain quantity.

Is the Supremum Property always true?

Yes, the Supremum Property is a fundamental property of real numbers and is always true. It is one of the axioms that defines the real number system and is accepted as a basic principle in mathematics.

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