We prove that the Jacobi iterative method converges from any initial values for solving the linear system resulting from a fourth-order compact finite difference discretization of the 2D convection-diffusion equation with constant convection coefficients. The convergence is assured regardless of the magnitude of the convection coefficients and the discretization mesh. This proof confirms the unconditional stability (convergence) of the 2D fourth-order compact scheme (with respect to classical iterative methods), a fact that has only been verified numerically but evaded rigorous justification for almost two decades.