Another Question on Upper and Lower Limits .... Denlinger, Theorem 2.9.6 (a)

In summary, the conversation revolves around understanding Theorem 2.9.6 (a) in Charles G. Denlinger's book "Elements of Real Analysis". The question asks for an explanation of why the inequality $\underline{x_n} \leq \underline{ x_{n + m} } \leq \overline{ x_{n + m} } \leq \overline{ x_m }$ implies that $\text{sup} \{ \underline{x_n} \ : \ n \in \mathbb{N} \} \leq \text{inf} \{ \overline{x_n} \ : \ n \in \mathbb{N} \}$. The conversation also
  • #1
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I am reading Charles G. Denlinger's book: "Elements of Real Analysis".

I am focused on Chapter 2: Sequences ... ...

I need help with the proof of Theorem 2.9.6 (a)Theorem 2.9.6 reads as follows:
View attachment 9250
View attachment 9251
In the above proof of part (a) we read the following:

" ... \(\displaystyle \forall \ m, n \in \mathbb{N}, \ \underline{x_n} \leq \underline{ x_{n + m} } \leq \overline{ x_{n + m} } \leq \overline{ x_m }\). Thus, \(\displaystyle \text{sup} \{ \underline{x_n} \ : \ n \in \mathbb{N} \} \leq \text{inf} \{ \overline{x_n} \ : \ n \in \mathbb{N} \}\)... ... "
My question is as follows:Can someone explain exactly why \(\displaystyle \ \underline{x_n} \leq \underline{ x_{n + m} } \leq \overline{ x_{n + m} } \leq \overline{ x_m } \ \) implies that \(\displaystyle \ \text{sup} \{ \underline{x_n} \ : \ n \in \mathbb{N} \} \leq \text{inf} \{ \overline{x_n} \ : \ n \in \mathbb{N} \}\) ... ... Hope that someone can help ...

Peter
===============================================================================

It may help MHB readers to have access to Denlinger's definitions and notation regarding upper and lower limits ... so I am providing access to the same ... as follows:
View attachment 9252
View attachment 9253
Hope that helps ...

Peter
 

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  • Denlinger - 1 - Start of Section 2.9  - PART 1 ... .png
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  • #2
Peter said:
Can someone explain exactly why \(\displaystyle \ \underline{x_n} \leq \underline{ x_{n + m} } \leq \overline{ x_{n + m} } \leq \overline{ x_m } \ \) implies that \(\displaystyle \ \text{sup} \{ \underline{x_n} \ : \ n \in \mathbb{N} \} \leq \text{inf} \{ \overline{x_n} \ : \ n \in \mathbb{N} \}\) ... ...
Leaving out the two intermediate parts in that string of inequalities, you see that $\underline{x_n} \leqslant \overline{ x_m }$ (for all $m$ and $n$ in $\Bbb{N}$). Keeping $m$ fixed and letting $n$ vary, it follows that $\overline{ x_m }$ is an upper bound for the sequence $\{ \underline{x_n}\}$. Therefore $\sup\{ \underline{x_n}\} \leqslant \overline{ x_m }$.

Now let $m$ vary. That last inequality shows that $\sup\{ \underline{x_n}\}$ is a lower bound for the sequence $\{\overline{ x_m }\}$, and therefore $\sup\{ \underline{x_n}\} \leqslant \inf \{\overline{ x_m }\}$.
 
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  • #3
Opalg said:
Leaving out the two intermediate parts in that string of inequalities, you see that $\underline{x_n} \leqslant \overline{ x_m }$ (for all $m$ and $n$ in $\Bbb{N}$). Keeping $m$ fixed and letting $n$ vary, it follows that $\overline{ x_m }$ is an upper bound for the sequence $\{ \underline{x_n}\}$. Therefore $\sup\{ \underline{x_n}\} \leqslant \overline{ x_m }$.

Now let $m$ vary. That last inequality shows that $\sup\{ \underline{x_n}\}$ is a lower bound for the sequence $\{\overline{ x_m }\}$, and therefore $\sup\{ \underline{x_n}\} \leqslant \inf \{\overline{ x_m }\}$.

Thanks Opalg ...

Appreciate your help ...

Peter
 

Related to Another Question on Upper and Lower Limits .... Denlinger, Theorem 2.9.6 (a)

1. What is the purpose of Denlinger's Theorem 2.9.6 (a)?

The purpose of Denlinger's Theorem 2.9.6 (a) is to provide a mathematical proof for the existence of upper and lower limits in a given set of numbers. This theorem is commonly used in statistical analysis and is essential in determining the behavior and characteristics of a data set.

2. How is Denlinger's Theorem 2.9.6 (a) applied in real-life scenarios?

Denlinger's Theorem 2.9.6 (a) is often used in various fields such as economics, finance, and engineering to analyze and predict trends in data. It is also used in quality control to determine the upper and lower limits of a product's specifications.

3. Can Denlinger's Theorem 2.9.6 (a) be used for any type of data?

Yes, Denlinger's Theorem 2.9.6 (a) can be applied to any type of data, whether it is numerical, categorical, or qualitative. The theorem is based on mathematical principles and does not depend on the type of data being analyzed.

4. Are there any limitations to Denlinger's Theorem 2.9.6 (a)?

Like any other mathematical theorem, Denlinger's Theorem 2.9.6 (a) has its limitations. It assumes that the data being analyzed is normally distributed and that there are no extreme outliers. If these assumptions are not met, the theorem may not accurately represent the data.

5. How does Denlinger's Theorem 2.9.6 (a) relate to other statistical concepts?

Denlinger's Theorem 2.9.6 (a) is closely related to other statistical concepts such as mean, median, and standard deviation. It is used to determine the upper and lower bounds for these measures of central tendency and variability. Additionally, this theorem is often used in conjunction with other statistical tests and methods to analyze data.

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