Recent experiments on N′-N-S junctions with high-temperature cuprate superconductors have shown that superconductivity can be induced in the adjacent, normal metal at distances many times greater than the few hundred Angstroms range of the conventional proximity effect. We show that this arises in the particular geometry used in these experiments through the constructive interference of the tails of the de Gennes-Saint-James bound and quasi-bound states of the normal metal N that penetrate into the third metal, N′. This constructive interference results in the re-appearance of a pair amplitude in this metal which is analogous to the “spin echo” of NMR and which we name a “pair echo.”