# The Number e .... Another Question Regarding Sohrab Proposition 2.3.15 ....

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In summary, the conversation discusses the proof of Proposition 2.3.15 in Houshang H. Sohrab's book "Basic Real Analysis" (Second Edition). The focus is on understanding why t_n \leq s_n \Longrightarrow \text{ lim sup } ( t_n ) \leq e ... ... , with the conversation providing an explanation using the definition of limsup.
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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 an another aspect of the proof of Proposition 2.3.15 ...

Proposition 2.3.15 and its proof read as follows:
View attachment 9072
In the above proof by Sohrab, we read the following:

" ... ... It follows that $$\displaystyle t_n \leq s_n$$ so that

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

My thoughts so far are as follows:

$$\displaystyle t_n \leq s_n$$$$\displaystyle \Longrightarrow \lim_{ n \to \infty } t_n \leq \lim_{ n \to \infty } s_n$$ $$\displaystyle \Longrightarrow \lim_{ n \to \infty } t_n \leq e$$But ... how/why can we conclude that $$\displaystyle \text{ lim sup } ( t_n ) \leq e$$ ... ... ?

***EDIT*** In the above thoughts I have wrongly assumed that we know, without further analysis, that $$\displaystyle (t_n)$$ is convergent ... Help will be appreciated ... ...

Peter

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Peter said:
Can someone please explain exactly how/why $$\displaystyle t_n \leq s_n \Longrightarrow \text{ lim sup } ( t_n ) \leq e$$ ... ... ?My thoughts so far are as follows:

$$\displaystyle t_n \leq s_n$$$$\displaystyle \Longrightarrow \lim_{ n \to \infty } t_n \leq \lim_{ n \to \infty } s_n$$ $$\displaystyle \Longrightarrow \lim_{ n \to \infty } t_n \leq e$$But ... how/why can we conclude that $$\displaystyle \text{ lim sup } ( t_n ) \leq e$$ ... ... ?
Slightly amend that to get

$$\displaystyle t_n \leqslant s_n$$

$$\displaystyle \Longrightarrow \limsup_{ n \to \infty } t_n \leqslant \limsup_{ n \to \infty } s_n$$

$$\displaystyle \Longrightarrow \limsup_{ n \to \infty } t_n \leqslant e$$

(because $\displaystyle\limsup_{ n \to \infty } s_n = \lim_{ n \to \infty } s_n = e$).

Opalg said:
Slightly amend that to get

$$\displaystyle t_n \leqslant s_n$$

$$\displaystyle \Longrightarrow \limsup_{ n \to \infty } t_n \leqslant \limsup_{ n \to \infty } s_n$$

$$\displaystyle \Longrightarrow \limsup_{ n \to \infty } t_n \leqslant e$$

(because $\displaystyle\limsup_{ n \to \infty } s_n = \lim_{ n \to \infty } s_n = e$).
Thanks Opalg ...

Reflecting on what you have written ...

Have to check things ... certainly did not know (could not find a Proposition) that $$\displaystyle t_n \leqslant s_n$$

$$\displaystyle \Longrightarrow \limsup_{ n \to \infty } t_n \leqslant \limsup_{ n \to \infty } s_n$$ ...Peter

Peter said:
Have to check things ... certainly did not know (could not find a Proposition) that $$\displaystyle t_n \leqslant s_n$$

$$\displaystyle \Longrightarrow \limsup_{ n \to \infty } t_n \leqslant \limsup_{ n \to \infty } s_n$$ ...
It's true, though! See if you can prove it yourself, using the definition of limsup.

Opalg said:
It's true, though! See if you can prove it yourself, using the definition of limsup.

Thanks for the help, Opalg ...

Yes, checked that out ... indeed basically follows from definition of lim sup ...

Thanks again

Peter

## 1. What is the number e?

The number e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828. It is an important number in mathematics and is often used in calculations involving exponential growth and decay.

## 2. Who discovered the number e?

The number e was first discovered by the Swiss mathematician Leonhard Euler in the 18th century. However, the concept of the number e was known to mathematicians before Euler, and it was also independently discovered by the Swiss mathematician Johann Bernoulli.

## 3. What is the significance of the number e?

The number e is significant because it appears in many natural phenomena, such as compound interest, population growth, and radioactive decay. It also has many important mathematical properties, making it a fundamental constant in calculus and other branches of mathematics.

## 4. How is the number e calculated?

The number e can be calculated in several ways, including using infinite series, continued fractions, and limits. One common way to calculate e is to use the infinite series: e = 1 + 1/1! + 1/2! + 1/3! + ...

## 5. What is Sohrab Proposition 2.3.15?

Sohrab Proposition 2.3.15 is a mathematical proposition from the book "The Number e: And Everything After" by Eli Maor. It states that the number e can be approximated by taking the sum of the first n terms of the infinite series 1 + 1/1! + 1/2! + 1/3! + ..., where n is a positive integer. This proposition is useful for calculating the value of e to a desired level of accuracy.

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