Sohrab ex. 2.1.12 part (1): ordering on the real numbers

[Math Amateur]

Homework Statement

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 2.1.12 Part (1) ... ...

Exercise 2.1.12 Part (1) reads as follows:

I am unable to make a meaningful start on Exercise 2.1.12 (1) ... can someone please help ...Relevant equations

Sohrab defines $\mathbb{R}$ as a field with binary operations of addition and multiplication ... he then goes on to define subtraction, division and exponentiation as follows:

Sohrab's definition of the usual ordering on $\mathbb{R}$ plus some of the properties following are as follows ... (but note that Exercises 2.1.10 and 2.1.11 precede Exercise 2.1.12 and so, I think, must be taken as given properties for the purposes of Exercise 2.1.12 ... ) ...

*** 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 \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 ...

3. The Attempt at a Solution

I am unable to make a meaningful start on this problem ... can someone help me to get started ...

Peter

 

[Math Amateur]

I think I have made some progress with showing that $\frac{a}{2} \gt 0$ ...We have $2 \gt 0$ (can we say this? why is it valid?)

and so $2^{-1} \gt 0$ by Exercise 2.1.11 (5) (see scanned text in above post)

So we now have

$a, 2^{-1} \in P$ (definition of a as greater than 0 )

$\Longrightarrow a \cdot 2^{-1} \in P$ (Order Axiom $O_2$ ... see scanned text in above post)

$\Longrightarrow \frac{a}{2} \gt 0$ (by definition of division ...see scanned text in above post)Is that correct?

If it is a valid and good proof ... then we still need to show $\frac{a}{2} \lt a$ ... but how ...?Peter

 

[fresh_42]

I think I have made some progress with showing that $\frac{a}{2} \gt 0$ ...
We have $2 \gt 0$ (can we say this? why is it valid?)

This is exercise 2.1.10 (c).

and so $2^{-1} \gt 0$ by Exercise 2.1.11 (5) (see scanned text in above post)

You can also use exercise 2.1.10 (c) again and show, that $\frac{1}{2}<0$ is impossible by $(O_2)$ and $(O_3)\,$.

So we now have
$a, 2^{-1} \in P$ (definition of a as greater than 0 )
$\Longrightarrow a \cdot 2^{-1} \in P$ (Order Axiom $O_2$ ... see scanned text in above post)
$\Longrightarrow \frac{a}{2} \gt 0$ (by definition of division ...see scanned text in above post)
Is that correct?

Yes.

If it is a valid and good proof ... then we still need to show $\frac{a}{2} \lt a$ ... but how ...?

You can use exercise 2.1.11 (1) and what you just have proven here.