  We show that relativistic quantum electrodynamics in the Coulomb gauge
satisfies the following bound, which establishes stability: let $H(\Lambda,V)$
denote the Hamiltonian of $QED_{1+3}$ on the three-dimensional torus of volume
$V$ and with ultraviolet cutoff $\Lambda$. Then there exists a constant
$0<\mu(\Lambda,V)<\infty$ (the vacuum energy renormalization) such that the
renormalized Hamiltonian is positive: $H_{ren}(\Lambda,V) \equiv
H_{\Lambda,V}+\mu_{\Lambda, V}\cdot \mathbb{1} \ge 0 $.
