N-particles approximation of the vlasov equations with singular potential

We prove the convergence in any time interval of a point-particle approximation of the Vlasov equation by particles initially equally separated for a force in 1/|x|^α, with α ≦ 1. We introduce discrete versions of the L ^∞ norm and time averages of the force-field. The core of the proof is to show that these quantities are bounded and that consequently the minimal distance between particles in the phase space is bounded from below.