
Ordered Fields
Alright so I'm not sure if this is the right thread for this, but this is the proof i need help with:
Let S be an ordered field and suppose x, y ∈ S. Using only the Ordered Field axioms, show that 0<x<y implies 0<(1/y) <(1/x).
I just need a starting point to go from but i'm stuck at defining that 1/y and 1/x are both elements of S

By definition of a field, $\displaystyle x^{1}$ and $\displaystyle y^{1}$ exist because $\displaystyle x,y$ are nonzero. What happens if you multiply all of $\displaystyle 0<x<y$ by $\displaystyle (xy)^{1}$?

You also need, of course, "if 0< x then 0< 1/x". Have you already proved that?