We consider a stochastic individual based model for a population structured by a vector trait and with logistic interactions. We consider its limit in a context from adaptive dynamics: the population is large, the mutations are rare and we view the process in the timescale of mutations. Using averaging techniques due to Kurtz (1992), we give a new proof of the convergence of the individual based model to the Trait substitution sequence of Metz et al. (1992) and rigorously proved by Champagnat (2006): assuming that ''invasion implies fixation'', we obtain in the limit a process that jumps from one population equilibrum to another one when mutations occur and invade the population.