Inequality involving metrics

Question

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.

Answer 1

If $\rho(x,z) \geq \rho(y,u)$ then $\rho(x,z) - \rho(y,u) \leq \rho(x,y) + \rho(y,z) - \rho(y,u) \leq \rho(x,y) + \rho(z,u)$. Where we have used $\rho(y,z) \leq \rho(y,u) + \rho(u,z)$ which implies $\rho(y,z) - \rho(y,u) \leq \rho(z,u)$ in the second inequality. The case $\rho(x,z) \leq \rho(y,u)$ is similar.