  We derive norm bounds that imply the convergence of perturbation theory in
fermionic quantum field theory if the propagator is summable and has a finite
Gram constant. These bounds are sufficient for an application in
renormalization group studies. Our proof is conceptually simple and technically
elementary; it clarifies how the applicability of Gram bounds with uniform
constants is related to positivity properties of matrices associated to the
procedure of taking connected parts of Gaussian convolutions. This positivity
is preserved in the decouplings that also preserve stability in the case of
two-body interactions.
