The Number e .... Sohrab Proposition 2.3.15 ....

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In summary, the conversation is about a request for help with understanding a specific part of the proof of Proposition 2.3.15 in Houshang H. Sohrab's book "Basic Real Analysis" (Second Edition). The person asking for help is wondering how and why the inequality ##t_n \leq s_n## leads to the result ##\limsup_n t_n \leq e##. Another person explains that this is due to the property that if ##t_n \leq s_n## for all ##n## (sufficiently large), then ##\limsup_n t_n \leq \limsup_n s_n##. They also mention that since ##\limsup_n s_n
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TL;DR Summary
Concerns a particular inequality in demonstrating the e = lim ( 1 + 1/n)^n ...
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 an aspect of the proof of Proposition 2.3.15 ...

Proposition 2.3.15 and its proof read as follows:
Sohrab - Proposition 2.3.15 ... .png

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

" ... ... It follows that ##t_n \leq s_n## so that

##\text{ lim sup } ( t_n ) \leq e## ... ... "
Can someone please explain exactly how/why ##t_n \leq s_n \Longrightarrow \text{ lim sup } ( t_n ) \leq e## ... ... ?



Help will be appreciated ... ...

Peter
 
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Recall that if we have ##t_n \leq s_n## for all ##n## (sufficiently large), then ##\limsup_n t_n \leq \limsup_n s_n##.

But, we know that ##\limsup_n s_n = \lim_n s_n = e##, hence the result.
 
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Yes ... can now see that ...

Thanks ...

Peter
 
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Related to The Number e .... Sohrab Proposition 2.3.15 ....

1. What is "The Number e" and why is it important in mathematics?

The Number e, also known as Euler's number, is a mathematical constant that is approximately equal to 2.71828. It is important because it appears in many mathematical formulas and has many applications in calculus, probability, and other branches of mathematics.

2. Who is Sohrab and what is Proposition 2.3.15?

Sohrab is not a person, but rather a mathematical notation used to represent a variable. Proposition 2.3.15 is a specific theorem or statement within a larger mathematical theory or proof.

3. Can you explain what Proposition 2.3.15 states?

Unfortunately, without further context or information, it is impossible to accurately explain what Proposition 2.3.15 states. It could refer to any number of different mathematical statements depending on the specific theory or proof it is a part of.

4. How is "The Number e" calculated?

The Number e is calculated using the infinite series: e = 1 + 1/1! + 1/2! + 1/3! + ... + 1/n!, where n approaches infinity. This means that e is the sum of 1 plus 1 divided by 1 factorial (1), plus 1 divided by 2 factorial (2), plus 1 divided by 3 factorial (6), and so on.

5. What are some real-world applications of "The Number e"?

The Number e has many real-world applications, including compound interest and continuous growth or decay. It is also used in probability and statistics to model continuous distributions. Additionally, it appears in many physical and scientific phenomena, such as radioactive decay and population growth.

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