  In this paper we show that there is a Lipatov bound for the radius of
convergence for superficially divergent one-particle irreducible Green
functions in a renormalizable quantum field theory if there is such a bound for
the superficially convergent ones. The radius of convergence turns out to be
${\rm min}\{\rho,1/b_1\}$, where $\rho$ is the bound on the convergent ones,
the instanton radius, and $b_1$ the first coefficient of the $\beta$-function.
