  We consider systems of $N$ particles in dimension one, driven by pair
Coulombian or gravitational interactions. When the number of particles goes to
infinity in the so called mean field scaling, we formally expect convergence
towards the Vlasov-Poisson equation. Actually a rigorous proof of that
convergence was given by Trocheris in \cite{Tro86}. Here we shall give a
simpler proof of this result, and explain why it implies the so-called
"Propagation of molecular chaos". More precisely, both results will be a direct
consequence of a weak-strong stability result on the one dimensional
Vlasov-Poisson equation that is interesting by it own. We also prove the
existence of global solutions to the $N$ particles dynamic starting from any
initial positions and velocities, and the existence of global solutions to the
Vlasov-Poisson equation starting from any measures with bounded first moment in
velocity.
