1. ## Ordered Fields, 0<a<b

I just read a proof that says 0 < a ==> 0 < a^-1.

How do I prove 0 < a < b ==> 0 < b^-1 < a^-1?

Also, the text did not give a definition for 0 < a < b.

Would it be correct to assume 0 < a and a < b <=> 0 < a < b, similarly for 0<= a and a<=b <=> 0 <= a <= b?

2. Originally Posted by Noxide
I just read a proof that says 0 < a ==> 0 < a^-1.

How do I prove 0 < a < b ==> 0 < b^-1 < a^-1?

Also, the text did not give a definition for 0 < a < b.

Would it be correct to assume 0 < a and a < b <=> 0 < a < b, similarly for 0<= a and a<=b <=> 0 <= a <= b?
Of course this last definition is the correct one. For the second part recall (c.f. Rudin) that one of the ordered field axioms is that $\displaystyle x<y$ and $\displaystyle z>0$ then $\displaystyle xz<yz$. Moreover, it's easy to show that $\displaystyle x,y>0\implies xy>0$. So you have that $\displaystyle 0<a<b$, $\displaystyle a^{-1},b^{-1}>0$ and thus from previous discussion $\displaystyle a^{-1}b^{-1}>0$ it follows that $\displaystyle 0<(a^{-1}b^{-1}}a<(a^{-1}b^{-1})b\cdots$

3. My text only gave me axioms for <=
Are < and <= equivalent ... I know they're not, but can one replace the other in the ordered field axioms?

4. Originally Posted by Noxide
My text only gave me axioms for <=
Are < and <= equivalent ... I know they're not, but can one replace the other in the ordered field axioms?
In all cases that immediately occur to me, yes.