"Simple" ordered fields proof: show that $0<y^{-1}<1$

Question

Given that $x>0$ and $y=x+1$ show that $0<y^{-1}<1$ specifying what proprieties of the Ordered Field you are using.

I really struggle to understand whether I have demonstrated the theorem properly or not.

Answer 1

Hint. Use the following facts:

• (Adding inequalities) $a\le b$ and $c\le d$ implies $a+c\le b+d$,
• If $0<a<b$, then $0<\frac1b<\frac1a$.