The objective of this paper is to find in a setting of n sequential observations of objects a good online policy to select the κ best of these n uniquely rankable objects. This focus is motivated by the fact that it is hard to find closed form solutions of optimal strategies for general κ and n. Selection is without recall, and the idea is to investigate threshold functions which maintain all present information, that is thresholds which are functions of all selections made so far. Our main interest lies in the asymptotic behaviour of these thresholds as n → ∞ and in the corresponding asymptotic performance of the threshold algorithm.