  We prove two equilibrium properties of a system of interacting atoms in three
or higher dimensional continuous space. (i) If the particles interact via pair
potentials of a nonnegative Fourier transform, their self-organization into
infinite permutation cycles is simultaneous with off-diagonal long-range order.
If the cycle lengths tend to infinity not slower than the square of the linear
extension of the system, there is also Bose-Einstein condensation. (ii) If the
pair potential is also nonnegative, cycles composed of a nonzero fraction of
the total number of particles do appear if the density exceeds a
temperature-dependent threshold value. The two together constitute the proof
that in such a system Bose-Einstein condensation takes place at high enough
densities.
