Limits of Sequences .... Bartle & Shebert, Example 3.4.3 (b)

[Math Amateur]

I am reading "Introduction to Real Analysis" (Fourth Edition) b Robert G Bartle and Donald R Sherbert ...

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

I need help in fully understanding Example 3.4.3 (b) ...

Example 3.4.3 (b) ... reads as follows:

In the above text from Bartle and Sherbert we read the following:

" ... ... Note that if $z_n := c^{ \frac{1}{n} }$ then $z_n \gt 1$ and $z_{ n+1 } \lt z_n$ for all $n \in \mathbb{N}$. (Why?) ... "Can someone help me to show rigorously that $z_n \gt 1$ and $z_{ n+1 } \lt z_n$ for all $n \in \mathbb{N}$ ... ... ?Hope that someone can help ...

Peter

 

[S.G. Janssens]

Suppose that $z^n \le 1$ for some $n \in \mathbb{N}$. Then $c = (z^n)^n \le 1^n = 1$, a contradiction.
Similarly, suppose that $z^n \le z^{n+1}$ for some $n \in \mathbb{N}$. Then
$$
\frac{z^n}{z^{n+1}} = c^{\frac{1}{n} - \frac{1}{n+1}} = c^{\frac{1}{n(n+1)}} \le 1,
$$
so $c \le 1^{n(n+1)} = 1$, also a contradiction.

Do you like the book?

 

[Math Amateur]

Suppose that $z^n \le 1$ for some $n \in \mathbb{N}$. Then $c = (z^n)^n \le 1^n = 1$, a contradiction.
Similarly, suppose that $z^n \le z^{n+1}$ for some $n \in \mathbb{N}$. Then
$$
\frac{z^n}{z^{n+1}} = c^{\frac{1}{n} - \frac{1}{n+1}} = c^{\frac{1}{n(n+1)}} \le 1,
$$
so $c \le 1^{n(n+1)} = 1$, also a contradiction.

Do you like the book?

Thanks for the help, Krylov ... just reflecting on what you have written ...

The book ... well ... I was looking for a rigorous text on one variable real analysis that covered the foundations of real analysis well and moderately thoroughly at about senior undergraduate level ... hopefully with detailed and complete proofs ... and I have found that Bartle and Sherbert meets my wishes pretty well ...

Two other books that I found met my need equally well (indeed perhaps better) are as follows:

" Basic Real Analysis" (Second Edition) ... 2014 ... by Houshang H Sohrab

and

"Real Analysis: Foundations and Functions of One Variable" (Fifth Edition) ... 2015 ... by Miklos Laczkovich and Vera T. Sos

Both books are well described on Amazon ...

Peter

 

[S.G. Janssens]

Thanks for the help, Krylov ... just reflecting on what you have written ...

Let me know if something in what I wrote is unclear.

and I have found that Bartle and Sherbert meets my wishes pretty well ...

Yes, I know this book and I think it is very good.I was curious what you would think of it.

Two other books that I found met my need equally well (indeed perhaps better) are as follows:

Thank you, I do not know these books but I am always interested in new (to me) titles in analysis.