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Order Fields Axioms

  1. Oct 4, 2012 #1
    1. The problem statement, all variables and given/known data
    Using only axioms o1-o4 and the result in part (i), show that for any a,b[itex]\in[/itex]F, with 0<a and 0<b, that if a[itex]^{2}[/itex]<[itex]^{2}[/itex]b, then a<b.


    2. Relevant equations
    o1) For any a,b[itex]\in[/itex]F, precisely one of the three following holds: a<b, b<a, a=b
    o2) If a,b,c[itex]\in[/itex]F, and if a<b and b<c, then a<c
    o3) For any a,b,c[itex]\in[/itex]F, if a<b, then a+c<b+c
    o4) For any a,b,c[itex]\in[/itex]F, if a<b and 0<c, then ac<bc

    Part (i): For any a,b,c,d[itex]\in[/itex]F, with 0<b and 0<c, if a<b and c<d, then ac<bd

    3. The attempt at a solution
    I have tried several dozen manipulations of this over the last several hours and it's just not coming together. I'm at the end of my rope with this one and I just want an answer. I'm well past the point where I would have a sense of accomplishment from figuring it out and I've still got more homework to finish by tomorrow morning, so any help with this would be greatly appreciated.
     
  2. jcsd
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