  We prove a $C^\infty$ version of the Nekhoroshev's estimate on the stability
times of the actions in close to integrable Hamiltonian systems. The proof we
give is a variant of the original Nekhoroshev's proof and it consists in first
conjugating, globally in the phase space, and up to a small remainder, the
system to a normal form. Then we perform the geometric part of the proof in the
normalized variables. As a result, we obtain a proof which is simpler than the
usual ones.
