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We consider a model of a population of fixed size $N$ undergoing selection.Each individual acquires beneficial mutations at rate $\mu_N$, and eachbeneficial mutation increases the individual's fitness by $s_N$. Eachindividual dies at rate one, and when a death occurs, an individual is chosenwith probability proportional to the individual's fitness to give birth. Undercertain conditions on the parameters $\mu_N$ and $s_N$, we obtain rigorousresults for the rate at which mutations accumulate in the population and thedistribution of the fitnesses of individuals in the population at a given time.Our results confirm predictions of Desai and Fisher (2007).

Rigorous results for a population model with selection I: evolution of the fitness distribution