  Two BPHZ convergence theorems are proved directly in Euclidean position
space, without exponentiating the propagators, making use of the Cluster
Convergence Theorem presented previously. The first theorem proves the absolute
convergence of arbitrary BPHZ-renormalized Feynman diagrams, when counterterms
are allowed for one-line-reducible subdiagrams, as well as for
one-line-irreducible subdiagrams. The second theorem proves the conditional
convergence of arbitrary BPHZ-renormalized Feynman diagrams, when counterterms
are allowed only for one-line-irreducible subdiagrams. Although the convergence
in this case is only conditional, there is only one natural way to approach the
limit, namely from propagators smoothly regularized at short distances, so that
the integrations by parts needed to reach an absolutely convergent integrand
can be carried out, without picking up short-distance surface terms. Neither
theorem requires translation invariance, but the second theorem assumes a much
weaker property, called "translation smoothness". Both theorems allow the
propagators in the counterterms to differ, at long distances, from the
propagators in the direct terms. For massless theories, this makes it possible
to eliminate all the long-distance divergences from the counterterms, without
altering the propagators in the direct terms. Massless theories can thus be
studied directly, without introducing a regulator mass and taking the limit as
it tends to zero, and without infra-red subtractions.
