  We consider a model for Darwinian evolution in an asexual population with a
large but non-constant populations size characterized by a natural birth rate,
a logistic death rate modelling competition and a probability of mutation at
each birth event. In the present paper, we study the long-term behavior of the
system in the limit of large population $(K\to \infty)$ size, rare mutations
$(u\to 0)$, and small mutational effects $(\sigma\to 0)$, proving convergence
to the canonical equation of adaptive dynamics (CEAD). In contrast to earlier
works, e.g. by Champagnat and M\'el\'eard, we take the three limits
simultaneously, i.e. $u=u_K$ and $\sigma=\sigma_K$, tend to zero with $K$,
subject to conditions that ensure that the time-scale of birth and death events
remains separated from that of successful mutational events. This slows down
the dynamics of the microscopic system and leads to serious technical
difficulties that requires the use of completely different methods. In
particular, we cannot use the law of large numbers on the diverging time needed
for fixation to approximate the stochastic system with the corresponding
deterministic one. To solve this problem we develop a "stochastic Euler scheme"
based on coupling arguments that allows to control the time evolution of the
stochastic system over time-scales that diverge with $K$.
