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Analysis problem: x>o -> 1/x > 0

  1. Feb 19, 2008 #1
    Analysis problem: x>o --> 1/x > 0

    1. The problem statement, all variables and given/known data
    Prove

    If x>0 --> 1/x > 0


    2. Relevant equations

    ordered field axioms

    closure, associativity, commutativity, identity, inverses, distributive law, trichotomy law, transitive law, preservation

    x+z = y+z --> x = y
    x.0 = 0
    -1.x = -x
    xy=0 iff x=0 or y=0
    x<y iff -y<-x
    x<y and z<0 then xz > yz


    3. The attempt at a solution

    i've tried this problem several times, and always hit a dead end

    i tried a direct proof

    assume x>0
    therefore x does not equal 0

    by existence of inverse

    there exists a unique 1/x such that x (1/x) = 1

    after that point, i get no where in my attempts

    any suggestions?

    you can user other "theorems" but you must also prove them
     
  2. jcsd
  3. Feb 19, 2008 #2

    morphism

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    Science Advisor
    Homework Helper

    Assume 1/x<0. Can you derive a contradiction using one of the properties you listed? (Hint: which one deals with things <0?)
     
  4. Feb 19, 2008 #3
    hey thanks a lot

    prove: x > 0 --> 1/x > 0

    assume 0 < x and assume 1/x less than or equal to 0, x cannot equal 0. then there exists a unique 1/x s.t. x(1/x) = 1. since 1/x is unique, 1/x cannot equal zero.
    therefore 1/x < 0

    since 0 < x and 1/x < 0

    0. 1/x > x . 1/x
    0 > 1

    which contradicts 0 < 1

    i would have to prove that 1 > 0, but i already done this.

    thanks
     
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