(b) Let . because . Suppose . Then , and . Hence , and the statement is true for all positive integers .
(e) Let is divisible by . Then is divisible by 5, so . Suppose . Then is divisible by 5. Hence is divisible by 5 because each term is. Hence , so and is divisible by 5 for every positive integer .
(b) If , then let and be the points in the th subinterval at which has its maximum and minimum, respectively. Then , by part (a), so that
(c) If , then as , . That means and have the same limit, so has area.