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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 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
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
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