The proof I am working on is:
Suppose $u$ and $v$ are two vertices in a graph $G$ with $$\epsilon (u):=\max_{v\in V} d(v,u)=m, \epsilon(v)=n$$ Prove $d(u, v) ≥ |m − n|$.
I know that the eccentricity is the max distance between a a vertex and other vertices, and I know the distance is the shortest path between two vertices, but I am struggling with how to begin this proof. I am not sure if I am missing a key piece of information, or how to connect them
Hint : Suppose $d(u,v)<|m-n|$. Take $z$ to be an arbitrary vertex of the graph and try to bound $d(u,z)$ for instance. Can you find a contradiction to $\epsilon(u) = m$, $\epsilon(v) = n$?