Ordering on the Set of Real Numbers .... Sohrab, Exercise 2.1.10 (c) ....

[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 Exercise 2.1.10 Part (c) ... ...

Exercise 2.1.10 Part (c) reads as follows:View attachment 7216I am unable to make a meaningful start on Exercise 2.1.10 (c) ... can someone please help ...

PeterNOTE 1: I am not sure what assumptions Sohrab wants us to make about \(\displaystyle \mathbb{N}, \mathbb{N_0}, \mathbb{Z}\) and \(\displaystyle \mathbb{Q}\) for these exercises ... he has not developed/constructed the natural numbers, the integers or the rationals ... but simply named them as sets and done little more than indicate notation for them ... as follows:View attachment 7217
https://www.physicsforums.com/attachments/7218So I am trying to prove the exercise using the real numbers as defined by an ordered field (which, I think, can be shown to contain a copy of each of the sets \(\displaystyle \mathbb{N}, \mathbb{N_0}, \mathbb{Z}\) and \(\displaystyle \mathbb{Q}\) ... )

Mind you ... after reading the start of Sohrab's Appendix A I am again a little uncertain as to what to assume when I read the following ... (mind you, this is stated well after the section where Exercise 2.1.10 (b) appears ... )View attachment 7219I would, however like, as I have previously indicated, to prove the exercise using the real numbers as defined by an ordered field ...

NOTE 2: Sohrab defines \(\displaystyle \mathbb{R} \) as a field with binary operations of addition and multiplication ... he then goes on to define subtraction, division and exponentiation as follows:View attachment 7220Sohrab's definition of the usual ordering on \(\displaystyle \mathbb{R}\) plus some of the properties following are as follows ...View attachment 7221
View attachment 7222Hope someone can help ...

Peter*** EDIT *** I am concerned that Exercises 2.1.1 and 2.1.2 contain properties of addition, multiplication and inverses that flow directly form the properties of \(\displaystyle \mathbb{R}\) as a field, ... ... and these properties could possibly be useful in the exercise ... so I am providing Sohrab's description of the field of real numbers and the exercises that follow it, namely Exercises 2.1.1 and 2.1.2 ... View attachment 7223
https://www.physicsforums.com/attachments/7224

 

[Euge]

Hint: Proof by induction and part (b) of the exercise.

 

[Math Amateur]

Hint: Proof by induction and part (b) of the exercise.

Thanks for the help, Euge ...

An attempt to follow your advice follows ...To show that \(\displaystyle n \in \mathbb{N} \Longrightarrow n \gt 0\) ... ... ... ... (1)Now ... the above statement (1) is true for \(\displaystyle n = 1\) by Exercise 2.1.10 (b)

Suppose now that it is true for some \(\displaystyle k \in \mathbb{N}\)...

Then \(\displaystyle k \in P\) ...

... but we have \(\displaystyle 1 \in P\)

Therefore \(\displaystyle k + 1 \in P\) ... ... ... by Order Axiom \(\displaystyle O_1\)

Therefore (1) is true for all \(\displaystyle n \in \mathbb{N}\)...Is that correct?

Peter

 

[solakis1]

Thanks for the help, Euge ...

An attempt to follow your advice follows ...To show that \(\displaystyle n \in \mathbb{N} \Longrightarrow n \gt 0\) ... ... ... ... (1)Now ... the above statement (1) is true for \(\displaystyle n = 1\) by Exercise 2.1.10 (b)

Suppose now that it is true for some \(\displaystyle k \in \mathbb{N}\)...

Then \(\displaystyle k \in P\) ...

... but we have \(\displaystyle 1 \in P\)

Therefore \(\displaystyle k + 1 \in P\) ... ... ... by Order Axiom \(\displaystyle O_1\)

Therefore (1) is true for all \(\displaystyle n \in \mathbb{N}\)...Is that correct?

Peter

Yes it is