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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 \(\displaystyle m \in \mathbb{N}\) and \(\displaystyle n \geq m\), we have

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

\(\displaystyle s_m = \sum_{ k = 0 }^m \frac{1}{ k! } \leq \text{ lim inf } (t_n)\) ... ... "

My question is as follows:

\(\displaystyle t_n \geq 1 + 1 + \frac{1}{ 2! } \left( 1 - \frac{1}{ n } \right) + \ ... \ + \frac{1}{ m! } \left( 1 - \frac{1}{ n } \right) \ ... \ \left( 1 - \frac{m - 1}{ n } \right) \) ... "

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 \(\displaystyle m \in \mathbb{N}\) and \(\displaystyle n \geq m\), we have

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

\(\displaystyle s_m = \sum_{ k = 0 }^m \frac{1}{ k! } \leq \text{ lim inf } (t_n)\) ... ... "

My question is as follows:

*" ... ... for any fixed \(\displaystyle m \in \mathbb{N}\) and \(\displaystyle n \geq m\), we have***How does the statement:**\(\displaystyle t_n \geq 1 + 1 + \frac{1}{ 2! } \left( 1 - \frac{1}{ n } \right) + \ ... \ + \frac{1}{ m! } \left( 1 - \frac{1}{ n } \right) \ ... \ \left( 1 - \frac{m - 1}{ n } \right) \) ... "

**and letting \(\displaystyle n \to \infty\) ...***... \(\displaystyle s_m = \sum_{ k = 0 }^m \frac{1}{ k! } \leq \text{ lim inf } (t_n)\) ... ... In other words ... can someone explain in some detail how/why \(\displaystyle s_m = \sum_{ k = 0 }^m \frac{1}{ k! } \leq \text{ lim inf } (t_n)\) ... ...is true?***... ... lead to the statement that**Help will be appreciated ...

Peter

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