We define a distance in $\displaystyle \mathbb{R}$ like the following:

$\displaystyle d(x,y) = \dfrac{|x-y|}{1+ |x-y|} $

A distance is defined by the following properties:

1. d(x,y) >= 0

2. d(x,y) = d(y,x)

3. d(x,y) =< d(x,z) + d(z,y)

I managed to prove the two first (it was rather easy). But I'm stuck on the third. Does anybody know how to do? I tried various things from expanding the first part to reducing the second part with the triangular inegality or even the inequation of Cauchy-Schwartz. I ended up always in something that I couldn't prove anymore.

I would be thankful for any idea.