Stability of QED

It is shown for a class of random, time-independent, square-integrable, three-dimensional magnetic fields that the one-loop effective fermion action of four-dimensional QED increases faster than a quadratic in $\mathbf{B}$ in the strong coupling limit. The limit is universal. The result relies on the paramagnetism of charged spin-$1/2$ fermions and the diamagnetism of charged scalar bosons.