1. The problem statement, all variables and given/known data If X is a metric space and A is a subset of X, show that [tex]|d(x,A) - d(y,A)|\leq d(x,y)[/tex] for any x,y in X. 2. Relevant equations Triangle inequality: [tex]|d(x,z) - d(y,z)|\leq d(x,y)[/tex], [tex]d(x,y)\leq d(x,z)+d(z,y)[/tex] 3. The attempt at a solution Fidling around with the triangle inequality. Edit: This is the third problem in G. Bredon's Topology and Geometry book. The goals of the exercice is to establish the continuity of the map [itex]x\mapsto d(x,A)[/itex], but she gives as a hint, "Use the triangle inequality to show that [itex]|d(x,A) - d(y,A)|\leq d(x,y)[/itex])" from which the result clearly follows in view of the continuity of the map [itex]x\mapsto d(x,x_0)[/itex] for some fixed x_0... The point of this edit is to bring attention to the exact wording of her hint, from which one can arguably conclude that the result should follow essentially from the triangle inequality.