We prove that a metric of constant scalar curvature on a polarised K\"ahler manifold is the limit of metrics induced from a specific sequence of projective embeddings; satisfying a condition introduced by H.~Luo. This gives, as a Corollary, the uniqueness of constant scalar curvature K\"ahler metrics in a given rational cohomology class. The proof uses results in the literature on the asymptotics of the Bergman kernel. The arguments are presented in a general framework involving moment maps for two different group actions.