We derive norm bounds that imply the convergence of perturbation theory in
fermionic quantum field theory if the propagator is summable and has a fini
te Gram constant. These bounds are sufficient for an application in renorma
lization group studies. Our proof is conceptually simple and technically el
ementary; it clarifies how the applicability of Gram bounds with uniform co
nstants is related to positivity properties of matrices associated to the p
rocedure of taking connected parts of Gaussian convolutions. This positivit
y is preserved in the decouplings that also preserve stability in the case
of two-body interactions.