In class we posed the following puzzle: $$Given n tasks indexed from $1$ to n$,$ where task i is characterized by a positive integer execution requirement e$\_$i and a positive integer weight w$\_$i $,$ show how to compute a $($single$-$resource, non$-$preemptive$)$ schedule for the n tasks that minimizes the total weighted completion time. Your algorithm should run in O$($n log n$)$ time.$$

An algorithm for solving this puzzle is as follows: Schedule the tasks in nonincreasing order of the ratio w$\_$i$/$e$\_$i $,$ breaking ties arbitrarily. Prove that this algorithm is correct.