The Number e .... Another Issue/Problem Regarding Sohrab Proposition 2.3.15 ....

[Math Amateur]

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 another issue/problem with the proof of Proposition 2.3.15 ...Proposition 2.3.15 and its proof read as follows:
View attachment 9076
In the above proof by Sohrab we read the following:

" ... ...Next, for any fixed $$m \in \mathbb{N}$$ and $$n \geq m$$, we have

$$t_n \geq 1 + 1 + \frac{1}{ 2! } \left( 1 - \frac{1}{ n } \right) + \ ... \ + \frac{1}{ m! } \left( 1 - \frac{1}{ n } \right) \ ... \ \left( 1 - \frac{m - 1}{ n } \right) $$so that letting $$n \to \infty$$ we get

$$s_m = \sum_{ k = 0 }^m \frac{1}{ k! } \leq \text{ lim inf } (t_n)$$ ... ... "
My question is as follows:

How does the statement:" ... ... for any fixed $$m \in \mathbb{N}$$ and $$n \geq m$$, we have

$$t_n \geq 1 + 1 + \frac{1}{ 2! } \left( 1 - \frac{1}{ n } \right) + \ ... \ + \frac{1}{ m! } \left( 1 - \frac{1}{ n } \right) \ ... \ \left( 1 - \frac{m - 1}{ n } \right) $$ ... " and letting $$n \to \infty$$ ...
... ... lead to the statement that ... $$s_m = \sum_{ k = 0 }^m \frac{1}{ k! } \leq \text{ lim inf } (t_n)$$ ... ... In other words ... can someone explain in some detail how/why $$s_m = \sum_{ k = 0 }^m \frac{1}{ k! } \leq \text{ lim inf } (t_n)$$ ... ...is true?

Help will be appreciated ...

Peter

 

[GJA]

Hi Peter,

can someone explain in some detail how/why $$s_m = \sum_{ k = 0 }^m \frac{1}{ k! } \leq \text{ lim inf } (t_n)$$ ... ...is true?

Take liminf on both sides of the inequality $$t_{n}\geq 1+1+\ldots$$ with respect to $n$. Since the limit of the right side with respect to $n$ exists and is equal to $s_{m}$, this is the value of the liminf of the right side.

 

[Math Amateur]

Hi Peter,
Take liminf on both sides of the inequality $$t_{n}\geq 1+1+\ldots$$ with respect to $n$. Since the limit of the right side with respect to $n$ exists and is equal to $s_{m}$, this is the value of the liminf of the right side.

Thanks GJA ...

Working on what you have suggested ...

Peter