We consider the problem of choosing a discounted-cost minimizing infinite-stage control sequence under nonstationary positive semidefinite quadratic costs and linear constraints. Specific cases include the nonstationary LQ tracker and regulator problems. We show that the optimal costs for finite-stage approximating problems converge to the optimal infinite-stage cost as the number of stages grows to infinity. Under a state reachability condition, we show that the set unions of all controls optimal to all feasible states for the finite-stage approximating problems converge to the set of infinite-stage optimal controls. A tie-breaking rule is provided that selects finite-stage optimal controls so as to force convergence to an infinite horizon optimal control.