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Spivak's calculus: is my proof wrong?

  1. Jul 12, 2010 #1
    1. The problem statement, all variables and given/known data
    If a, b > 0, and a2 < b2, then a < b.


    2. Relevant equations
    P is the collection of all positive numbers, n > 0

    (P10) Trichotomy law: For every number a, one and only one of the following holds:
    (i) a = 0
    (ii) a is in the collection P
    (iii) -a is in the collection P

    (P11) If a and b are in P, then a + b are in P
    (P12) If a and b are in P, then a x b is in P.


    3. The attempt at a solution

    If a2 < b2, then 0 < b2 - a2, hence b2 - a2 is in the collection P

    0 < b2 - a2 which is the same as 0 < (b - a)(b + a)

    by (P12), (b - a) and (b + a) are both in P,

    if (b - a) is in P, this means b - a > 0, hence, b > a

    this isn't the answer in the book, I will post that after dinner, but my proof seems sound but at the same time I can't tell if it's correct or wrong.
     
  2. jcsd
  3. Jul 12, 2010 #2

    Hurkyl

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    That's not P12. The statement you used is
    If a x b is in P, then a and b are in P​
    which is the converse of P12, and is only true half the time.
     
  4. Jul 12, 2010 #3
    I got it. If ab is in P then it's possible a and b are not in P, since it can be said: a < 0, b < 0 ,ab > 0 (product of two negatives). I missed that. Thanks.
     
  5. Aug 18, 2011 #4
    But can I say that if (b-a)(b+a) > 0, either (b-a) and (b+a) have to be positive, or either negative, and, as I know by hypothesis that (b+a) >= 0, (b-a) must be in P, hence (b-a) > 0, so b > a?
     
  6. Aug 18, 2011 #5

    micromass

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    You will need to prove this...
    Otherwise it's correct.

     
  7. Aug 18, 2011 #6
    Sure, but that is the easy part hehe. Thanks!
     
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