Past exam of the mathematics course of the University of Cambridge / 2015 / ia / Paper 1 / 5B / iii / Solution

Apply the triangle inequality to the path from $x_1$ through $y_2$ to $y_1$:This is one direction of the reverse triangle inequality. A second application of the triangle inequality, this time through $x_2$, givesThe resulting four-point triangle inequality follows without any assumptions on coincidences among the four points.