INTRODUCTORY EXERCISES

(2) Using the basic rules BR1 - BR12, show that the following property of real numbers holds:

 If a > 0, then 1/a > 0. 

We are assuming a > 0. This implies ¹/_a exists (BR-8) and ¹/_a 0 (otherwise, (¹/_a)a would be 0, not 1). Thus, by BR-10, either ¹/_a > 0 or -¹/_a > 0. We now show that -¹/_a > 0 is impossible. If -¹/_a > 0 were true, then, since a > 0 also, we must have

                                 (-1/a)a > 0           BR-12.

But (-¹/_a)a = -1, so this contradicts 1 > 0. Thus, we must conclude that ¹/_a > 0.