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

[Math Amateur]

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:

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

 

[member 587159]

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.

 

[Math Amateur]

Yes ... can now see that ...

Thanks ...

Peter