I've proved that if $(X, d)$ is a geodesic metric space then there exists a graph which is quasi-isometric to $X$...during this proof I've precisely used the fact that given two point in $X$ there exists a distance minimizing curve inside $X$ joining those two points...

After this I've tried to generalize my proof for arbitrary metric space...which I could not able to prove till now...Now I think it is not true in general...I've an intuitive guess for a counter-example which is ...consider $X=\{x_i \mid i \in \mathbb{N}\}$ and $d(x_i,x_j)= |i^2 - j^2|$. I think for this metric space there dose not exists any graph quasi-isometric with ($X,d$)...but till now I could not able to prove it.

So can any-body provide me some hints or some counter-example for this fact...or may be some proof of the fact that given any metric space there exists a graph quasi-isometric to that space.

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