The temperature evolution of the proximity effect in Au/La$_{2-x}$Sr$_x$CuO$_4$ and La$_{1.55}$Sr$_{0.45}$CuO$_4$/La$_{2-x}$Sr$_x$CuO$_4$ bilayers was investigated using scanning tunneling microscopy. Proximity induced gaps, centered at the chemical potential, were found to persist above the superconducting transition temperature, $T_c$, and up to nearly the pseudogap crossover temperature in both systems. Such independence of the spectra on the details of the normal metal cap layer is incompatible with a density-wave order origin. However, our results can be accounted for by a penetration of incoherent Cooper pairs into the normal metal above $T_c$.
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