  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 $T_c$.
