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Math Help - Having trouble formulating proof...

  1. #1
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    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
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  2. #2
    Junior Member
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    Nov 2008
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    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.
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