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 , thus . Since n is odd, , thus , which means that an odd power of x is negative.
Thus the sign is preserved for odd n.
I am completely new to proofs, so please bare with me.
I'm simply trying to prove that if
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 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 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