  We show, by explicit computation, that bare lattice perturbation theory in
the two-dimensional O(n) nonlinear $\sigma$ models with superinstanton boundary
conditions is divergent in the limit of an infinite number of points
$|\Lambda|$. This is the analogue of David's statement that renormalized
perturbation theory of these models is infrared divergent in the limit where
the physical size of the box tends to infinity. We also give arguments which
support the validity of the bare perturbative expansion of short-distance
quantities obtained by taking the limit $|\Lambda|\to\infty$ term by term in
the theory with more conventional boundary conditions such as Dirichlet,
periodic, and free.
