(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that x < y and n is odd implies x^n < y^n

This is from chapter 1 problem 6 (b), so I am sure a lot of you are familiar with this. One can only use Spivak's 12 axioms, theorems derived from said axioms, and mathematical induction.

2. Relevant equations

x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + ... + xy^{n-2} + y^{n-1})

Sorry I do not know how the scripting language works here.

3. The attempt at a solution

I have tried doing this problem for about five hours, *shame*, and I have looked up hints on the internet about it but have been still unsuccessful. I think my problem with this exercise is my accounting.

Here is one of my attempts:

If x < y [itex]\wedge[/itex] n is odd [itex]\Rightarrow[/itex] x^n < y^n

There are three cases to consider:

(1) 0 [itex]\leq[/itex] x < y

(2) x < 0 [itex]\leq[/itex] y

(3) x < y [itex]\leq[/itex] 0

Case (1) is a special case of the previous theorem we proved in problem 6 (a), so it's trivial.

Case (2) is a negative number to an odd power is less than a positive number to an odd power, so again trivial. But since I am so new to proofs, it would be nice if one of you nice gentlemen or ladies would help me show this. (I am about 1/4 the way done with the book How To Prove It and my school has never taught me anything about proofs in my 1 and 1/2 years of attendance)

Case (3) is the case which I have been stuck on.

x < y [itex]\leq[/itex] 0

x - y < 0 < -y

∴ x - y < 0

Notice all three cases can produce this inequality so there must be one logical way to prove them all at once WLOG, but I do not see it?

I've also tried to prove this other ways, namely induction and contrapositive, but they did not get me any closer. Thanks for your time, and I really appreciate any and all responses.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Spivak's Calculus problem 6

**Physics Forums | Science Articles, Homework Help, Discussion**