I have an Approximation Algorithm problem and its answer but I am having a hard time to clearly understand. The problem is as follows:Consider the following closest-point heuristic for building an approximate traveling-salesman tour whose cost function satisfies the triangle inequality. Begin with a trivial cycle consisting of a single arbitrarily chosen vertex. At each step, identify the vertex $u$ that is not on the cycle but whose distance to any vertex on the cycle is minimum. Suppose that the vertex on the cycle that is nearest $u$ is vertex $v$. Extend the cycle to include $u$ by inserting $u$ just after $v$. Repeat until all vertices are on the cycle. Prove that this heuristic returns a tour whose total cost is not more than twice the cost of an optimal tour.The answer is (I apoligize for including an image but I couldnt get the  some special characters displayed correctly)What I do not understand is the part surrounded by the red rectangle. I will very much appreciate if you help me to get the idea behind it. I am studying for an exam and I am stuck with this problem.Respectfully,Jorge Maldonado
Closest-point heuristic approximation problem
approximation