  We show, within the framework of the Euclidean $\phi^4$-quantum field theory
in four dimensions, that the Wilson operator product expansion (OPE) is not
only an asymptotic expansion at short distances as previously believed, but
even converges at arbitrary finite distances. Our proof rests on a detailed
estimation of the remainder term in the OPE, of an arbitrary product of
composite fields, inserted as usual into a correlation function with further
"spectator fields". The estimates are obtained using a suitably adapted version
of the method of renormalization group flow equations. Convergence follows
because the remainder is seen to become arbitrarily small as the OPE is carried
out to sufficiently high order, i.e. to operators of sufficiently high
dimension. Our results hold for arbitrary, but finite, loop orders. As an
interesting side-result of our estimates, we can also prove that the "gradient
expansion" of the effective action is convergent.
