Changing interest rates and present values

Let's relax the assumption that short-term nominal interest rates are constant. Let i_t denote the nominal interest rate in year t; i.e., a dollar invested in a one-year bond in year t - 1 returns 1 + i_t dollars in year t. Assume that nominal interest rates are positive. Consider a consol that promises a coupon payment of C dollars each year. Let PV_t denote the present value of the consol in year t. Derive an expression for PV_t in terms of C and future interest rates (i_t+i, i_t+2, ...). Use the product notation introduced in the first problem set: If z_1, z_2, ... z_n is a list of numbers, then we write z_m to mean z_1 times z_2 times ... times z_n. Suppose that is increasing: i_t+i > i_t. Show that PV_t is decreasing: PV_i+1