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