On the terminal condition for the Bellman equation for dynamic optimization with an infinite horizon

In this work a sufficient condition for deterministic dynamic optimization with discrete time and infinite horizon is formulated. It encompasses also situations where the instantaneous payoff/utility function can attain infinite values.The usual terminal condition for sufficiency of the Bellman equation requiring that the limit superior of the value function along each admissible trajectory is equal to 0 is replaced by a weaker one in which the limit superior of the value function can attain nonpositive values.This kind of terminal condition is applicable also to deterministic dynamic optimization problems with real-valued instantaneous payoff function in which the usual terminal condition does not hold.