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Math Help - Ordered Fields

  1. #1
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    Ordered Fields

    Alright so I'm not sure if this is the right thread for this, but this is the proof i need help with:
    Let S be an ordered field and suppose x, y ∈ S. Using only the Ordered Field axioms, show that 0<x<y implies 0<(1/y) <(1/x).

    I just need a starting point to go from but i'm stuck at defining that 1/y and 1/x are both elements of S
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    By definition of a field, x^{-1} and y^{-1} exist because x,y are nonzero. What happens if you multiply all of 0<x<y by (xy)^{-1}?
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  3. #3
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    You also need, of course, "if 0< x then 0< 1/x". Have you already proved that?
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