Prove proposition on real numbers and inverses.

Question

Prove the following proposition

Let $x, y \in \mathbb{ R}>0$. If $x < y$ then $0 < y^{-1 }< x^{-1}.$

So far I've gotten that since $x, y > 0$ then $x^{-1}, y^{-1} > 0$.

Answer 1

Since $x,y>0$ and hence $xy>0$ you can multiply $x<y$ with $(xy)^{-1}$ and get what you wanted

Answer 2

Hint: $1/y-1/x=x/xy-y/xy$. Do you see?