Systematic evolution of ligands by exponential enrichment (SELEX) is a procedure by which a mixture of nucleic acids that vary in sequence can be separated into pure components with the goal of isolating those with specific biochemical activities. The basic idea is to combine the mixture with a specific target molecule and then separate the target-NA complex from the resulting reaction. The target-NA complex is then separated by mechanical means (for example by filtration), the NA is then eluted from the complex, amplified by polymerase chain reaction (PCR) and the process repeated. After several rounds, one should be left with a pool of [NA] that consists mostly of the species in the original pool that best binds to the target. In Irvine et al. [Irvine, D., Tuerk, C., Gold, L., 1991. SELEXION, systematic evolution of nucleic acids by exponential enrichment with integrated optimization by non-linear analysis. J. Mol. Biol. 222, 739-761] a mathematical analysis of this process was given. In this paper we revisit Irvine et al. [Ibid]. By rewriting the equations for the SELEX process, we considerably reduce the labor of computing the round to round distribution of nucleic acid fractions. We also establish necessary and sufficient conditions for the SELEX process to converge to a pool consisting solely of the best binding nucleic acid to a fixed target in a manner that maximizes the percentage of bound target. The assumption is that there is a single nucleic acid binding site on the target that permits occupation by not more than one nucleic acid. We analyze the case for which there is no background loss (no support losses and no free [NA] left on the support). We then examine the case in which such there are such losses. The significance of the analysis is that it suggests an experimental approach for the SELEX process as defined in Irvine et al. [Ibid] to converge to a pool consisting of a single best binding nucleic acid without recourse to any a priori information about the nature of the binding constants or the distribution of the individual nucleic acid fragments.