OPM 1996 Problem 2

Prove that given three positive numbers, we can choose two of them, say $x$ and $y,$ with $x >y$ such that $$\frac{x-y}{1 +xy }<1.$$ Prove also that if the number $1$ that appears in the second member of the previous inequality is replaced by a lower number, even if very close to $1$ , the previous proposition is false.