  We consider a one dimensional periodic forward-backward parabolic equation,
regularized by a non-linear fourth order term of order $\epsilon^2\ll 1$. This
equation is known in the literature as Cahn-Hilliard equation with degenerate
mobility. Under the hypothesis of the initial data being well prepared, we
prove that as $\epsilon\to0$, the solution converges to the solution of a
well-posed degenerate parabolic equation. The proof exploits the gradient flow
nature of the equation in $\mathcal{W}^2$ and utilizes the framework of
convergence of gradient flows developed by Sandier-Serfaty. As an incidental,
we study fine energetic properties of solutions to the Thin-film equation
$\partial_t\nu=(\nu\nu_{xxx})_x$.
