  We study counterterms (CT's), candidates for UV divergences in the
four-dimensional N=8 supergravity. They have been constructed long ago in a
Lorentz covariant on shell superspace and recently in the chiral light-cone
(LC) superspace. We prove that all of these CT's are ruled out since they are
not available in the real LC superspace. This implies the perturbative UV
finiteness of d=4 N=8 supergravity under the assumption that supersymmetry and
continuous E7 symmetry are anomaly-free. The proof, based on the chiral nature
of CT's in the LC superspace, is a generalization of the perturbative F-term
non-renormalization theorem for N=8 supergravity.
