Upper Confidence Trees (UCT) are now a well known algorithm for sequential decision making; it is a provably consistent variant of Monte-Carlo Tree Search. However, the consistency is only proved in a the case where both the action space is finite. We here propose a proof in the case of fully observable Markov Decision Processes with bounded horizon, possibly including infinitely many states and infinite action spaces and arbitrary stochastic transition kernels. We illustrate the consistency on two benchmark problems, one being a legacy toy problem, the other a more challenging one, the famous energy unit commitment problem.