Prove this inequality involving metrics

[TMO]

For any metric $(X, \rho)$ and points therein, prove that $|\rho(x, z) - \rho(y, u)| \leq \rho(x, y) + \rho(z, u)$.

I know that this will involve iterated applications of the triangle inequality...but I still need another hint on how to proceed.

 

[mfb]

Did you draw a sketch? Consider distances to z-x+y, for example.