ABSTRACT: When it is known that the sequence of Henstock integrals of functions fn converges to the Henstock integral of a function f we give necessary and sufficient conditions for the sequence of integrals of the product fngn to converge to the integral of fg for all convergent sequences gn of functions of uniform bounded variation. The conditions are easy to apply and involve either the uniform boundedness or uniform convergence of the indefinite integrals of fn. The proof uses Stieltjes integrals and applies to bounded or unbounded intervals on the real line. It is shown how to define Stieltjes integrals on unbounded intervals without treating them as improper integrals. The Abel and Dirichlet tests for integrability of a product are obtained as corollaries as well as a form of the Riemann-Lebesgue lemma.