# Thread: Having trouble formulating proof...

1. ## Having trouble formulating proof...

I am completely new to proofs, so please bare with me.

I'm simply trying to prove that if $x\,<\,y\,\, then \,\,x^n\,<\,y^n\,\,for\,all\,odd\, n$

This seems simple enough, and i understand why it's true, but i'm having trouble expressing it in the form of a proof. I have that if x is greater than zero, then $x^n$ is also greater than zero. This is also true for all x greater than or less than zero provided that the number is raised to an even power. I reach this by arguing that the expression can always be reduced to factors of $x^2$ provided n is even, and thus the product is a product of positive numbers only. I see that the sign is preserved for odd n, which is the concept which underlies this proof, but i'm having trouble expressing it mathematically.

Could anyone help? Is there a good guide online somewhere regarding the formulation of basic proofs? The text I am working from is Spivak's Calculus (a great read) and I am able to follow his proofs, but this is the first time i've had to construct them myself.

Cheers,

John

2. To show that the sign is preserved for odd n, try this:

if x>0, then clearly any power of x is also >0.

if x<0, then $x=-|x|$, thus $x^n=(-1)^n|x|$. Since n is odd, $(-1)^n=-1$, thus $x^n=-|x|$, which means that an odd power of x is negative.

Thus the sign is preserved for odd n.