The workload formulation due to Harrison and coworkers of multiclass queueing networks has been fundamental to its analysis. Until recently, there was no actual theory which started with the physical queue and showed that under heavy traffic conditions, the optimal costs could be approximated by those for an optimization problem using the "limit" workload equations. Recently, this was done via viscosity solution methods by Martins, Shreve, and Soner for one important class. For this same class of problems (and including the cases not treated there), we use weak convergence methods to show that the sequence of optimal costs for the original network converges to the optimal cost for the workload limit problem. The proof is simpler and allows weaker (and non-Markovian) conditions. It uses current techniques in weak convergence analysis. It seems to be the first analysis of such multiclass "workload" problems by weak convergence methods. The general structure of the development seems applicable to the analysis of more complex systems.