Let $d: \mathbb{C}\times\mathbb{C} \to \mathbb{C}$ defined by
$d(x,y) = \frac{|x-y|}{\sqrt{1+|x|^2}+\sqrt{1+|y|^2}}$
All the properties of $d$ can easily be satisfied.
I proved that $d$ is a bounded metric while trying to solve this. By
$\frac{|x-y|}{\sqrt{1+|x|^2}+\sqrt{1+|y|^2}} \le \frac{|x|+|y|}{|x|+|y|}=1$
But I can't find a way to prove or disprove the triangle inequality. Any help will be much appreciated. Thanks.