  We consider a system of N trapped bosons with repulsive interactions in a
combined semiclassical mean-field limit at positive temperature. We show that
the free energy is well approximated by the minimum of the Hartree free energy
functional - a natural extension of the Hartree energy functional to positive
temperatures. The Hartree free energy functional converges in the same limit to
a semiclassical free energy functional, and we show that the system displays
Bose-Einstein condensation if and only if it occurs in the semiclassical free
energy functional. This allows us to show that for weak coupling the critical
temperature decreases due to the repulsive interactions.
