Reinforcement Learning (RL) is a methodology used to solve Markov decision processes (MDPs) within simulators. In the classical Actor-Critic (AC), a popular RL algorithm, the values of the so-called actor become unbounded. A recently introduced variant of the AC keeps the actor's values naturally bounded. However, the algorithm's convergence properties have not been established mathematically in the literature. Numerically, the bounded AC was studied under the Boltzmann action-selection strategy, but not under the more popular ϵ-greedy strategy in which the probability of selecting any non-greedy action converges to zero in the limit. The paper revisits the AC framework. A short review of the existing literature in the growing field of ACs is first presented. Thereafter, the algorithm is investigated for its convergence properties, under ϵ-greedy action selection, numerically on a small-scale MDP, as well as mathematically via the ordinary differential equation framework.