- #1

CaptainAmerica17

- 59

- 10

## Homework Statement

I'm currently working through Spivak independently and have reached the problems at the end of ch. 1.

The problem is:

Prove that if [itex] 0 < a < b [/itex], then [tex] a < \sqrt{ab} < \frac{a+b}{2} < b [/tex]

## Homework Equations

Spivak's properties P1 - P12

## The Attempt at a Solution

I was thinking that if I do it by cases, I can first try to prove [itex] a < \sqrt{ab} [/itex] and then do [itex] \frac{a+b}{2} < b[/itex]. After that, I can use [itex] 0 < a < b [/itex] to bring it all together. To start with [itex] a < \sqrt{ab} [/itex], I have: If [itex] a > 0 [/itex] and [itex] b > 0 [/itex], then [itex] ab > 0 [/itex]. I'm not sure where to go from there or if my approach is even correct.