Constrained Markov decision processes (CMDPs) are a common way to model safety constraints in reinforcement learning. State-of-the-art methods for efficiently solving CMDPs are based on primal-dual algorithms. For these algorithms, all currently known regret bounds allow for error cancellations --- one can compensate for a constraint violation in one round with a strict constraint satisfaction in another. This makes the online learning process unsafe since it only guarantees safety for the final (mixture) policy but not during learning. As \citet{efroni2020exploration} pointed out, it is an open question whether primal-dual algorithms can provably achieve sublinear regret if we do not allow error cancellations. In this paper, we give the first affirmative answer. We first generalize a result on last-iterate convergence of regularized primal-dual schemes to CMDPs with multiple constraints. Building upon this insight, we propose a model-based primal-dual algorithm to learn in an unknown CMDP. We prove that our algorithm achieves sublinear regret without error cancellations.