Ordered field. Please point me in the right direction.

[Kingdu57]

If a,b are elements of an ordered field F, and 0<a<b, how do I prove 0< 1/b < 1/a using the properties of an ordered field?

This seems so obvious to me but putting together a proof has proven very difficult.

 

[pka]

If a,b are elements of an ordered field F, and 0<a<b, how do I prove 0< 1/b < 1/a using the properties of an ordered field?

Any proof strictly depends upon the exact statements of the axioms and sequence of theorems you have to use. We do not have access to those.

But in general a proof goes something like this.
If \(\displaystyle x>0\) then \(\displaystyle \dfrac{1}{x}>0.\)
If \(\displaystyle x>0~y>0\) then \(\displaystyle xy>0\).

So multiply both sides of \(\displaystyle 0<a<b\) by \(\displaystyle \dfrac{1}{ab}.\)