- #1

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I am reading Charles G. Denlinger's book: "Elements of Real Analysis".

I am focused on Chapter 2: Sequences ... ...

I need help with the proof of Theorem 2.9.6 (a)

Theorem 2.9.6 reads as follows:

View attachment 9250

View attachment 9251

In the above proof of part (a) we read the following:

" ... \(\displaystyle \forall \ m, n \in \mathbb{N}, \ \underline{x_n} \leq \underline{ x_{n + m} } \leq \overline{ x_{n + m} } \leq \overline{ x_m }\). Thus, \(\displaystyle \text{sup} \{ \underline{x_n} \ : \ n \in \mathbb{N} \} \leq \text{inf} \{ \overline{x_n} \ : \ n \in \mathbb{N} \}\)... ... "

My question is as follows:

Can someone explain exactly why \(\displaystyle \ \underline{x_n} \leq \underline{ x_{n + m} } \leq \overline{ x_{n + m} } \leq \overline{ x_m } \ \) implies that \(\displaystyle \ \text{sup} \{ \underline{x_n} \ : \ n \in \mathbb{N} \} \leq \text{inf} \{ \overline{x_n} \ : \ n \in \mathbb{N} \}\) ... ...

Hope that someone can help ...

Peter

===============================================================================

It may help MHB readers to have access to Denlinger's definitions and notation regarding upper and lower limits ... so I am providing access to the same ... as follows:

View attachment 9252

View attachment 9253

Hope that helps ...

Peter

I am focused on Chapter 2: Sequences ... ...

I need help with the proof of Theorem 2.9.6 (a)

Theorem 2.9.6 reads as follows:

View attachment 9250

View attachment 9251

In the above proof of part (a) we read the following:

" ... \(\displaystyle \forall \ m, n \in \mathbb{N}, \ \underline{x_n} \leq \underline{ x_{n + m} } \leq \overline{ x_{n + m} } \leq \overline{ x_m }\). Thus, \(\displaystyle \text{sup} \{ \underline{x_n} \ : \ n \in \mathbb{N} \} \leq \text{inf} \{ \overline{x_n} \ : \ n \in \mathbb{N} \}\)... ... "

My question is as follows:

Can someone explain exactly why \(\displaystyle \ \underline{x_n} \leq \underline{ x_{n + m} } \leq \overline{ x_{n + m} } \leq \overline{ x_m } \ \) implies that \(\displaystyle \ \text{sup} \{ \underline{x_n} \ : \ n \in \mathbb{N} \} \leq \text{inf} \{ \overline{x_n} \ : \ n \in \mathbb{N} \}\) ... ...

Hope that someone can help ...

Peter

===============================================================================

It may help MHB readers to have access to Denlinger's definitions and notation regarding upper and lower limits ... so I am providing access to the same ... as follows:

View attachment 9252

View attachment 9253

Hope that helps ...

Peter

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