Let S be a set of positive real numbers. If S contains at least four distinct elements show that there are elements x,y ∈ S such that

$Given\ z(x,\ y) = \dfrac{x- y}{1 + xy}\ and\ x,\ y \in S,$

$\text{where S is a set of at least 4 distinct positive real numbers}$

$Prove\ \exists\ x,\ y \in\ S\ such\ that\ 0 < z(x,\ y) < \dfrac{\sqrt{3}}{3}.$

What can you say if S has at least 7 elements? What if S has at least n elements, where n > 2?