  In this paper we establish best approximation property of fully discrete
Galerkin solutions of second order parabolic problems on convex polygonal and
polyhedral domains in the $L^\infty(I;W^{1,\infty}(\Om))$ norm. The
discretization method consists of continuous Lagrange finite elements in space
and discontinuous Galerkin methods of arbitrary order in time. The method of
the proof differs from the established fully discrete error estimate techniques
and uses only elliptic results and discrete maximal parabolic regularity for
discontinuous Galerkin methods established by the authors in
\cite{LeykekhmanD_VexlerB_2016b}. In addition, the proof does not require any
relationship between spatial mesh sizes and time steps. We also establish
interior best approximation property that shows more local dependence of the
error at a point.
