Suppose $(X,d)$ is a metric space. We define a second metric $\psi: X \times X \to [0, \infty)$ given by
$$\psi(x,y) = \frac{d(x,y)}{d(x,y)+1}$$
I would like to prove in particular that $\psi$ satsifies the triangle inequality. But it seems trickier than it would first appear. So far I've said
Let $x,y,z \in X$. Since $d$ is a metric, we know that
$$d(x,z) \leq d(x,y) + d(y,z)$$
My next step is to convert $d(x,z)$ into $\psi(x,z)$. So I divide everything by $d(x,z) +1$ and obtain
$$\psi(x,z) = \frac{d(x,z)}{d(x,z)+1} \leq \frac{d(x,y)}{d(x,z)+1} + \frac{d(y,z)}{d(x,z)+1}$$
At this point though, I stop. I'm not certain this is the proper way to approach this proof. Any suggestions?
