Consider the following closest$-$point heuristic for building an approximate traveling salesman tour whose cost function satisfies the triangle inequality. Start by selecting an arbitrary city v$.$ Assume that this city is a valid cycle $($into itself$)$ representing the traveling salesman path. Find the city u that is closest to city v and add it to the cycle by inserting city u immediately after city v$.$ Now you have a cycle through two cities $($v $$ u$).$ Iteratively continue by finding the city u that is closest to any city in the existing cycle. Let$$s say that the closest city from the cycle is city v$.$ Again, add u immediately after v and repeat this until you have no cities left outside the cycle. QUESTIONS: $1.$ Provethat this heuristic returns a tour whose total cost is not more than twice the cost of an optimal tour. Take into account the triangle inequality.