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Following a recent study by our group in which a large $T_c$ enhancement wasreported in bilayers of the non-superconducting $La_{1.65}Sr_{0.35}CuO_4$ andsuperconducting $La_{1.88}Sr_{0.12}CuO_4$ films [Phys. Rev. Lett. \textbf{101},057005 (2008)], we checked if a similar effect occurs when superconducting$La_{1.88}Sr_{0.12}CuO_4$ islands are coated with a continuous layer of the nonsuperconducting $La_{1.65}Sr_{0.35}CuO_4$. We found that no such phenomenon isobserved. The bare superconducting islands film behaves as an insulator wheretransport occurs via hopping or tunneling between islands, but exhibits a weaksignature of localized superconductivity at $\sim$18 K. When over coated with a$La_{1.65}Sr_{0.35}CuO_4$ film, it becomes superconducting with a $T_c$ onsetof 16.6 K, which is less than that of a thick $La_{1.88}Sr_{0.12}CuO_4$ film(25.7 K). We therefore conclude that the lower $T_c$ in the bilayer is due to aconventional proximity effect.

Conventional proximity effect in bilayers of superconducting underdoped  $La_{1.88}Sr_{0.12}CuO_4$ islands coated with non superconducting overdoped  $La_{1.65}Sr_{0.35}CuO_4$