  We show that general cutoff scalar field theories in four dimensions are
perturbatively renormalizable through the use of diagrammatic techniques and an
adapted BPH renormalization method. Weinberg's convergence theorem is used to
show that operators in the Lagrangian with dimension greater than four, which
are divided by powers of the cutoff, produce perturbatively only local
divergences in the two-, three-, and four-point correlation functions. We also
show that the renormalized Green's functions are the same as in ordinary
$\Phi^4$ theory up to corrections suppressed by inverse powers of the cutoff.
These conclusions are consistent with those of existing proofs based on the
renormalization group.
