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

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SUMMARY

The discussion centers on Proposition 2.3.15 from Houshang H. Sohrab's "Basic Real Analysis" (Second Edition), specifically regarding the proof that establishes the relationship between sequences \( t_n \) and \( s_n \). It is concluded that if \( t_n \leq s_n \) for sufficiently large \( n \), then \( \limsup_{n} t_n \leq \limsup_{n} s_n \). Given that \( \limsup_{n} s_n = \lim_{n} s_n = e \), it follows that \( \limsup_{n} t_n \leq e \). This logical progression clarifies the proof's validity.

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  • Familiarity with sequences and series of real numbers
  • Knowledge of the properties of limits, specifically lim sup
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  • Study the concept of lim sup in detail, focusing on its properties and applications
  • Review Chapter 2 of "Basic Real Analysis" by Houshang H. Sohrab for deeper insights
  • Explore examples of sequences where \( t_n \leq s_n \) to solidify understanding
  • Investigate other propositions in real analysis that utilize similar limit properties
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TL;DR
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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