Suppose a greedy algorithm is used to return the smallest number of coins for a total of n cents, using coin denominations of 1, 2, and 4 cents. The algorithm selects one coin at a time, dispenses this coin, decreases n by the denomination of this coin, and proceeds in this fashion until n reaches zero. The greedy choice consists in always selecting a coin with the largest possible value. The required coins of given denominations are always available. Prove that this greedy algorithm always returns an optimal result.