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The modulation with magnetic field of the sheet inductance measured on proximity effect Josephson-junction arrays (JJAs) is progressively vanishing on lowering the temperature, leading to a low-temperature field-independent response. This behaviour is consistent with the decrease of the two-dimensional penetration length below the lattice parameter. Low-temperature data are quantitatively compared with theoretical predictions based on the XY model in the absence of thermal fluctuations. The results show that the description of a JJA within the XY model is incomplete and the system is put well beyond the weak screening limit which is usually assumed in order to invoke the well-known frustrated XY model describing classical Josephson-junction arrays. Copyright (C) EPLA, 2009