  Adaptive dynamics so far has been put on a rigorous footing only for clonal
inheritance. We extend this to sexually reproducing diploids, although
admittedly still under the restriction of an unstructured population with
Lotka-Volterra-like dynamics and single locus genetics (as in Kimura's 1965
infinite allele model). We prove under the usual smoothness assumptions,
starting from a stochastic birth and death process model, that, when
advantageous mutations are rare and mutational steps are not too large, the
population behaves on the mutational time scale (the 'long' time scale of the
literature on the genetical foundations of ESS theory) as a jump process moving
between homozygous states (the trait substitution sequence of the adaptive
dynamics literature). Essential technical ingredients are a rigorous estimate
for the probability of invasion in a dynamic diploid population, a rigorous,
geometric singular perturbation theory based, invasion implies substitution
theorem, and the use of the Skorohod $M_1$ topology to arrive at a functional
convergence result. In the small mutational steps limit this process in turn
gives rise to a differential equation in allele or in phenotype space of a type
referred to in the adaptive dynamics literature as 'canonical equation'.
