Suppose X is a δ-hyperbolic metric space and α is a geodesic in X. Let P : X → α denote any nearest point projection map, i.e. for all x ∈ X, d(x, α) = d(x, P(x)). Show that the P coarsely L-Lipschitz where L is a constant depending only on δ.
My attempt is: Considering the geodesic line joining P(z) and P(z'), And geodesic joining z and z', denoted [z,z']. I choose the midpoint p on [z,z'] and consider the geodesic triangle with vertices P(z), P(z') and p. Then, I am looking at the geodesic quadrilateral formed by P(z), P(z'), z and z'. Using the δ Hyperbolicity [P(z),P(z')] I want to express the L- Lipschitz condition but unable to do so. Any help is appreciated.