Abstract: A compact symplectic manifold endowed with a prequantum line bundle L can be embedded in the projective space by certain sections of high p-tensor powers of L. We show that the Fubini-Study metrics induced by these embeddings converge at speed rate 1/p^2 to the symplectic form. This should give a generalization to the almost Kähler case of Donaldson's lower bounds of the Calabi functional.