Proving x<y x^n<y^n if n is Odd

[Ronnin]

Homework Statement

x<y x^{n}<y^{n} if n is odd

Homework Equations

The Attempt at a Solution

I am trying to prove this. I understand that if n were even it would eliminate any negatives for negative values of x or y, but how would I go about writing the proof for this. Would I break it down into different cases? Could I define what odd number is and just somehow state that because of that there would not be a possible sign change for x or y. I'm new to proofs so translating some of the most ovious stuff into mathmatical terms is tricky for me at times.

 

[micromass]

I would break it to pieces. Consider 0<x<y, x<0<y and x<y<0.

Alternatively, you could also proceed by induction, if you know this technique...

 

[Ronnin]

Have not started using induction yet. I am still muddling along with these basic direct proofs. Thanks for the suggestion.

 

[Ronnin]

This problem has been giving me hell. I have used 2k+1 for n and have tried to factor it out giving this (x^{k}x^{1/2}+y^{k}x^{1/2})(x^{k}x^{1/2}-y^{k}x^{1/2}) which is just one big circle because this only yeilds a \sqrt{x^{n}}= \sqrt{y^{n}}. I have even tried using the difference of cubes to attempt to factor this out with no luck. Could someone drop a hint as to whether I'm way off beat here or I'm missing a logical detail that would let me make some kind of algebraic magic happen. Thanks!

 

[micromass]

You could maybe start by proving: If x and y are positive and x<y, then xⁿ<yⁿ.
You can do this by induction.
This would show the result for positive numbers. If one of the numbers is negative, then you could perhaps use the equality (-x)ⁿ=-xⁿ if n is odd...

 

[Ronnin]

Since this is from Spivak chapter 1 I don't think I can use induction. I'm new to proofs and have never used that technique.

 

[sponsoredwalk]

Honestly if you study the chapter deeply enough you'll see that the way to prove this
example is given to you. One thing I was really really confused about with Spivak in
this chapter was his explanation of the Trichotomy Law, P10, & didn't really get it
until I read "An Introduction to Inequalities" by Bellman. I would seriously advise reading
that little book to learn how to deal with some of the proofs in chapter 1 of Spivak first but if
you can't do that then yeah the material is contained in the chapter & the trichotomy
axiom is a big hint as are the responses in this thread