  In this paper, we give a new proof of the classical KAM theorem on the
persistence of an invariant quasi-periodic torus, whose frequency vector
satisfies the Bruno-R\"ussmann condition, in real-analytic non-degenerate
Hamiltonian systems close to integrable. The proof, which uses rational
approximations instead of small divisors estimates, is an adaptation to the
Hamiltonian setting of the method we introduced in a previous work for
perturbations of constant vector fields on the torus.
