Let $(E,d)$ be a metric space and assume that the metric d satisfies $d(x,z) \leq \max(d(x,y),d(y,z))$ for all $x,y,z\in E$.
Prove that if $d(x,y) \neq d(y,z)$ then $d(x,z)=\max(d(x,y),d(y,z))$.
I have spent literally hours on this and I feel like a moron because I can't figure it out. Please help! Thanks!