Calculate diameter of metric space

Question

Calculate the diameter of space $ \left( \mathbb{R} , d \right) $ , where $ d : \mathbb{R} × \mathbb{R} \to \mathbb{R} $ is defined $ d(x,y) = \left|\left(\frac{x}{1+ \sqrt{ 1 + x^2}} - \frac{y}{1+ \sqrt{1 + y^2}}\right)\right| $ I don't even know how to start.

Answer 1

Hint: Note that \begin{align} \left|\frac{x}{1+\sqrt{1+x^2}}-\frac{y}{1+\sqrt{1+y^2}} \right| \leq \frac{|x|}{1+\sqrt{1+x^2}}+\frac{|y|}{1+\sqrt{1+y^2}} \leq 2 \end{align}