Supremum Property (AoC), Archimedean Property, Nested Intervals Theorem ....

In summary, the conversation discusses Theorem 2.1.45 from Houshang H. Sohrab's book "Basic Real Analysis" (Second Edition), which concerns the Supremum Property, the Archimedean Property, and the Nested Intervals Theorem. The conversation includes questions about the proof of the theorem, specifically regarding the location of $\sup(S)$ and the intersection of $I_n$ and $S$. The expert summarizer explains that the proof involves narrowing down the location of $\sup(S)$ through an approximation process and clarifies that $I_n \cap S \ne \emptyset$ because there is an element of $S$ that is greater than $s + \frac{k_n-1
  • #1
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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
 
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  • #2
Peter said:
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 \(\displaystyle s + \frac{m}{ 2^n} = \text{Sup}(S)\) ... ... ?
No. What is happening in this proof is that you narrow down the location of $\sup(S)$ by an approximation process. You start from a point $s$ in $S$, and having fixed $n$, you look at the points $s + \frac1{2^n}, s + \frac2{2^n}, s + \frac3{2^n}, \ldots$, until you find the first point $s + \frac{k_n}{2^n}$ that is an upper bound for $S$. That way, you have narrowed down the location of $\sup(S)$ to an interval $I_n$ of length $\frac1{2^n}$. You then increase $n$ to $n+1$, so as to locate $\sup(S)$ within a smaller interval $I_{n+1}$ (which must be either the first half or the second half of the previous interval $I_n$).

Peter said:
Question 2

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 \(\displaystyle I_n \cap S \ne \emptyset\)?
Since $k_n$ is the smallest integer for which $s + \frac{k_n}{2^n}$ is an upper bound for $S$, it follows that $s + \frac{k_n-1}{2^n}$ is not an upper bound for $S$. Therefore there is an element, call it $t$, of $S$ that is greater than $s + \frac{k_n-1}{2^n}$. But $t\leqslant s + \frac{k_n}{2^n}$ (because $s + \frac{k_n}{2^n}$ is an upper bound for $S$), and so $t$ lies in the interval $I_n$. So $t$ lies in the intersection $I_n\cap S$.
 
  • #3
Opalg said:
No. What is happening in this proof is that you narrow down the location of $\sup(S)$ by an approximation process. You start from a point $s$ in $S$, and having fixed $n$, you look at the points $s + \frac1{2^n}, s + \frac2{2^n}, s + \frac3{2^n}, \ldots$, until you find the first point $s + \frac{k_n}{2^n}$ that is an upper bound for $S$. That way, you have narrowed down the location of $\sup(S)$ to an interval $I_n$ of length $\frac1{2^n}$. You then increase $n$ to $n+1$, so as to locate $\sup(S)$ within a smaller interval $I_{n+1}$ (which must be either the first half or the second half of the previous interval $I_n$).Since $k_n$ is the smallest integer for which $s + \frac{k_n}{2^n}$ is an upper bound for $S$, it follows that $s + \frac{k_n-1}{2^n}$ is not an upper bound for $S$. Therefore there is an element, call it $t$, of $S$ that is greater than $s + \frac{k_n-1}{2^n}$. But $t\leqslant s + \frac{k_n}{2^n}$ (because $s + \frac{k_n}{2^n}$ is an upper bound for $S$), and so $t$ lies in the interval $I_n$. So $t$ lies in the intersection $I_n\cap S$.
Thanks Opalg ... most helpful ...

... indeed, you made it very clear ...

Peter
 

FAQ: Supremum Property (AoC), Archimedean Property, Nested Intervals Theorem ....

What is the Supremum Property?

The Supremum Property, also known as the Axiom of Completeness (AoC), is a fundamental concept in real analysis that states every non-empty set of real numbers that is bounded above has a least upper bound, or supremum. In other words, the supremum is the smallest number that is greater than or equal to every element in the set.

What is the Archimedean Property?

The Archimedean Property states that for any two positive real numbers, there exists a natural number n such that n times the first number is greater than the second number. In other words, the real numbers are "dense" in the sense that there are always infinitely many numbers between any two numbers.

What is the Nested Intervals Theorem?

The Nested Intervals Theorem is a theorem in real analysis that states if a nested sequence of closed intervals has a common point, then the intersection of all the intervals is non-empty. In other words, if each interval in the sequence contains the next interval, then there exists a real number that is contained in all of the intervals.

Why are these properties important in real analysis?

These properties are important in real analysis because they provide a solid foundation for understanding the real numbers and their properties. They allow us to make precise statements about real numbers and their relationships, and they are essential in proving many important theorems in calculus and other areas of mathematics.

How are these properties used in practical applications?

The Supremum Property, Archimedean Property, and Nested Intervals Theorem have many practical applications in fields such as physics, engineering, and economics. For example, the Supremum Property is used in optimization problems to find the maximum or minimum value of a function. The Archimedean Property is used in calculus to define the concept of a limit, and the Nested Intervals Theorem is used in computer algorithms for root-finding and numerical integration.

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