The expected rank of the individual selected in the secretary problem can be kept bounded as the number of individuals becomes infinite even if we are not permitted to remember more than one individual at a time. The least upper bound of these expected ranks is derived: it is approximately 7.4.
"The Finite-Memory Secretary Problem." Ann. Probab. 5 (4) 627 - 635, August, 1977. https://doi.org/10.1214/aop/1176995774