Showing Order Fields Axioms Hold: a<b for 0<a,b in F

[hammonjj]

Homework Statement

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.

Homework 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

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.

 

[mighty2000]

I understand your frustration and I am happy to help you with this problem. First, let's rewrite the given statement using the given axioms:

If a^{2}<^{2}b, then a<b

Now, let's start by assuming that a^{2}<^{2}b and try to prove that a<b. Using o1, we know that either a<b, b<a, or a=b. Let's consider each case separately.

Case 1: a<b
Since we are assuming that a^{2}<^{2}b, we can use o4 to get a^{2}c<^{2}bc. Then, using o2 and our assumption that a<b, we can conclude that a^{2}c<^{2}bc<^{2}b^{2}. Since c is positive, we can use o4 again to get ac<b^{2}c. And finally, using o2 and our assumption that a<b, we can conclude that ac<b^{2}c<b^{2}b=b^{3}. This shows that ac<b^{3}, which is equivalent to a<b.

Case 2: b<a
Using a similar approach to case 1, we can show that b<a is not possible.

Case 3: a=b
Since a^{2}<^{2}b, we can use o4 to get a^{2}c<^{2}bc. But since a=b, we know that a^{2}=b^{2}. So we can rewrite this as b^{2}c<b^{2}c. This is a contradiction, so our assumption that a=b must be false.

Therefore, we have shown that if a^{2}<^{2}b, then a<b. I hope this helps and good luck with the rest of your homework.