Consider introducing compound interest to the pricing formulas for perpetuities and annuities. Suppose each annual payment C is paid in n installments, spread equally over each year, and let r denote the nominal annual interest rate.

(a) (10) Show that the present value of a perpetuity does not depend on the number of compounding periods.

(b) (10) Show that the present value of an annuity is increasing in the number of compounding periods. What if the payments are made continuously throughout the year?