Bose-Einstein condensation in a harmonic trap:: Effect of interactions on Tc

For a harmonically trapped dilute Bose gas with uniformly repulsive interactions which is assumed to satisfy a certain condition on the extensivity of fluctuations, I find on upper bound on the condensate fraction f. If BEC is defined by the condition that f > const. N(-alpha), alpha < 1/ 2, I argue that in the limit N --> infinity, V(o) --> 0, NV(o) --> const. where V(o) is the space integral of the potential, the interactions cannot increase the critical temperature over that of the noninteracting gas.