  In this paper, we provide a much simplified proof of the main result in [Lin,
Xu, Zhang, arXiv:1302.5877] concerning the global existence and uniqueness of
smooth solutions to the Cauchy problem for a 2D incompressible viscous and
non-resistive MHD system under the assumption that the initial data are close
to some equilibrium states. Beside the classical energy method, the
interpolating inequalities and the algebraic structure of the equations coming
from the incompressibility of the fluid are crucial in our arguments. We
combine the energy estimates with the $L^\infty$ estimates for time slices to
deduce the key $L^1$ in time estimates. The latter is responsible for the
global in time existence.
