# Having trouble formulating proof...

• Nov 25th 2008, 11:49 AM
pico
Having trouble formulating proof...
I am completely new to proofs, so please bare with me.

I'm simply trying to prove that if \$\displaystyle 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 \$\displaystyle 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 \$\displaystyle 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
• Nov 26th 2008, 07:56 AM
JD-Styles
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 \$\displaystyle x=-|x|\$, thus \$\displaystyle x^n=(-1)^n|x|\$. Since n is odd, \$\displaystyle (-1)^n=-1\$, thus \$\displaystyle x^n=-|x|\$, which means that an odd power of x is negative.

Thus the sign is preserved for odd n.