We study the process of fixation of beneficial mutations in an asexual population by means of a theoretical model. Particularly, we wish to investigate how the supply of deleterious and beneficial mutations influences the dynamics of the adaptive process of an evolving population. It is well known that the deleterious mutations drastically affect the fate of beneficial mutations. In addition, an increasing supply of favorable mutations, to compensate the decay of the fitness due to the accumulation of deleterious mutations, produces the clonal interference phenomenon where advantageous mutations in distinct lineages compete to reach fixation. This competition imposes a limit to the speed of adaptation of the population. Intuitively, we would expect that the interplay of the two mechanisms would conspire to ensure fixation of only large-effect beneficial mutations. Our results, however, show that beneficial mutations of small effect have an increased probability of fixation when both beneficial and deleterious mutations rates are increased.