Minima of initial segments of infinite sequences of reals

Suppose that 〈xk〉k∈ℕ is a countable sequence of real numbers. Working in the usual subsystems for reverse mathematics, RCA0 suffices to prove the existence of a sequence of reals 〈uk〉k∈ℕ such that for each k, uk is the minimum of {x0, x1, …, xk}. However, if we wish to prove the existence of a sequence of integer indices of minima of initial segments of 〈xk〉k∈ℕ, the stronger subsystem WKL0 is required. Following the presentation of these reverse mathematics results, we will derive computability theoretic corollaries and use them to illustrate a distinction between computable analysis and constructive analysis. (© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)