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Spivak Calc: Is my proof correct?

  1. Sep 24, 2011 #1
    1. The problem statement, all variables and given/known data
    Prologue Chapter 1. Problem 8.


    (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

    P10: Let (b-a) be any number d. Then P'10 reduces to
    i) d = 0
    ii) 0 <d
    iii) 0 >d
    This is in essence what P10 is, except with the variable being different

    P11: Let there be numbers a, b, c, d.
    If a<b and c<d, then by P'12 the following are true:
    1) a+c < b+c
    2) b+c < b+d
    By P'11, a + c < b + d
    Looking at the case where 0 < b and 0 < d, b + d > 0 which is P11.

    P12: If 0 < a and 0 < c, by P'13, 0 < ac, which is P12
  2. jcsd
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