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Direct Air Capture (DAC) is a crucial technology for climate change mitigation. Adsorption-based DAC has been demonstrated at various scales, but suffers from sorbent degradation, high energy consumption and high costs. We propose a hybrid adsorption-membrane process where adsorption and membranes operate in a loop. The CO2-enriched stream produced by adsorption is further purified by membranes, and the almost pure N2 retentate from membranes is used as desorption purge. Therefore, the process integration allows regenerating the sorbent with N2 purge, a regeneration mode that is well-known to improve sorbent stability but that has never been considered feasible for real-world applications because of the high costs and the low CO2 purity produced. By adding a downstream membrane process based on pyridinic-graphene membranes, we achieve high CO2 purity, and we abate the costs for the purge by recirculating the retentate. The optimized energy consumption of the hybrid process is comparable with those of stand-alone adsorption technologies within the same productivity range. By considering an extended lifetime in the case of N2 purge, based on experimental data from the literature, we demonstrate the economic competitiveness of the hybrid process and its robustness towards multiple uncertainties in costs and grid emission intensity.