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Transferring learning-based models to the real world remains one of the hardest problems in model-free control theory. Due to the cost of data collection on a real robot and the limited sample efficiency of Deep Reinforcement Learning algorithms, models are usually trained in a simulator which theoretically provides an infinite amount of data. Despite offering unbounded trial and error runs, the reality gap between simulation and the physical world brings little guarantee about the policy behavior in real operation. Depending on the problem, expensive real fine-tuning and/or a complex domain randomization strategy may be required to produce a relevant policy. In this paper, a Soft-Actor Critic (SAC) training strategy using incremental environment complexity is proposed to drastically reduce the need for additional training in the real world. The application addressed is depth-based mapless navigation, where a mobile robot should reach a given waypoint in a cluttered environment with no prior mapping information. Experimental results in simulated and real environments are presented to assess quantitatively the efficiency of the proposed approach, which demonstrated a success rate twice higher than a naive strategy.