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1.21 Proof Rudin Principles of Mathematical Analysis - Uniqueness of the n-roots in R

  1. help! rudin 1.21

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  2. help! rudin 1.21

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  1. May 28, 2012 #1
    Hi!

    I need some help here, please.

    In Principles of Mathematical Analysis, Rudin prove the uniqueness of the n-root on the real numbers. For doing so, he prove that both $y^n < 0$ and that $y^n > 0$ lead to a contradiction.

    In the first part of the proof, he chooses a value $h$ such that
    1. $0<h<1$
    2 $h<\frac{x-y^n}{n(y+1)^(n-1)}$

    My question is: how does he know such a value $h$ exists?

    I know this value is positive, but I am not sure how to prove is less than 1.

    To add some context:
    x is a real positive number
    n is a integer positive number
    y is a real positive number

    If someone can help me with this, I will be very grateful!

    Thanks
     
  2. jcsd
  3. May 28, 2012 #2
    Re: 1.21 Proof Rudin Principles of Mathematical Analysis - Uniqueness of the n-roots

    Even if it is not less than 1 , you still can choose h less than this fraction and in the same time 0<h<1.
     
  4. May 28, 2012 #3
    Re: 1.21 Proof Rudin Principles of Mathematical Analysis - Uniqueness of the n-roots

    Don't get it. How can it be not less than 1, and at the same time, less than one?

    I think the more likely answers are that:

    1) No need to prove the existence of h
    2) There is something I'm missing about y=sup E, where E is the set all positive real numbers t, such that t^n<x

    By the way, perhaps I am not writing correctly the equations so that they show clearly in latex style.
     
  5. May 28, 2012 #4
    Re: 1.21 Proof Rudin Principles of Mathematical Analysis - Uniqueness of the n-roots

    I have Rudin at my hand now, so no source of confusion. My understanding is the following:
    Given any F>0 (which is the fraction you wrote above), it is possible to choose h such that :
    0 < h < 1 < F (in case 1<F)
    or
    0 < h < F <1 (in case F<1)

    In other words all what you need is to make sure that the fraction is greater than zero which is actually the case here.
     
  6. May 28, 2012 #5
    Re: 1.21 Proof Rudin Principles of Mathematical Analysis - Uniqueness of the n-roots

    Thanks. I am trying to get it now. But the fraction I wrote above is h, NOT x.
     
  7. May 28, 2012 #6
    Re: 1.21 Proof Rudin Principles of Mathematical Analysis - Uniqueness of the n-roots

    The fraction is NEITHER x NOR h. I edited my post and renamed it F as I realized calling it x is a bad idea.

    F= (x-yn)/n(y+1)n-1
    and F>0
     
  8. May 28, 2012 #7
    Re: 1.21 Proof Rudin Principles of Mathematical Analysis - Uniqueness of the n-roots

    Thank you.
     
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