  We consider the relative entropy between vacuum states of two different
theories: a conformal field theory (CFT), and the CFT perturbed by a relevant
operator. By restricting both states to the null Cauchy surface in the causal
domain of a sphere, we make the relative entropy equal to the difference of
entanglement entropies. As a result, this difference has the positivity and
monotonicity properties of relative entropy. From this it follows a simple
alternative proof of the c-theorem in d=2 space-time dimensions and, for d>2,
the proof that the coefficient of the area term in the entanglement entropy
decreases along the renormalization group (RG) flow between fixed points. We
comment on the regimes of convergence of relative entropy, depending on the
space-time dimensions and the conformal dimension $\Delta$ of the perturbation
that triggers the RG flow.
