Zakon: How can natural elements be uniquely represented in an ordered field?

[dawoodvora]

I am learning mathematics on my own, self-study type. Currently following Zakon's first book in his 3 part series.

In chapter 2, section 6, I have been successfully able to solve all the problem till I encountered 11'. It seems simple enough, but I am unable to understand the problem in the first place. I believe, Once I understand the problem itself, I will be in a position to chalk out a solution.

Please help me out in actually understanding the problem.


Here is the problem:

Chapter 2, Section 6, Problem 11’

11. Show by induction that each natural element x of an ordered field F can be uniquely represented as X=n ∙1', where n is a natural number in E¹ (n ∈N) and 1' is the unity in F; that is, x is the sum of n unities.

Conversely, show that each such n ∙1' is a natural element of F.

Finally, show that, for m,n ∈N, we have

m<n iff mx<nx, provided x>0

 

[micromass]

The question asks you to prove that every natural element x can be written as

$$x=1+1+1+1+...+1$$

where we sum 1 a number of times. Conversely, (this is very easy) every element of the form

$$1+1+1+1+1+...+1$$

belongs to F.