# Order Fields Axioms

1. Oct 4, 2012

### hammonjj

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$\in$F, with 0<a and 0<b, that if a$^{2}$<$^{2}$b, then a<b.

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

Part (i): For any a,b,c,d$\in$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.