Why does the product 1* × (-1*) equal all of Q in Dedekind cuts?

[Artusartos]

Homework Statement

For number 6.4 in this link:

http://people.ischool.berkeley.edu/~johnsonb/Welcome_files/104/104hw2sum06.pdf

I don't understand why the product of 1* and -1* gives us all of Q...so can anybody please explain that to me?

Thanks in advance

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

The cut 1* is the set of all rational numbers less than 1. The cut -1* is the set of all rational numbers less than -1. Their product is the set containing all products of numbers in those sets.

Let q be any positive rational number, let r be any negative rational number, less than -1 and let s= q/r, also a negative rational number. Then r is in -1*, s is in 1* and so q is in (1)*(-1)*. Thus all positive numbers are in this cut. Since if a is in a cut and if b< a, then b is also in the cut, it follows that all rational numbers are in it.

(I presume that the point of this exercise is to show why we cannot define the product of two cuts in this way.)