A formal proof of the convergence of the numerical escalator boxcar train (EBT) method is presented. EBT is commonly used in theoretical biology for the investigation of the dynamics of physiologically structured population models. A framework of measure-valued solutions is used for the analysis of the method and it is shown that the sequence of approximating solution measures generated by the EBT method converges weakly to the true solution measure under weak conditions on the growth rate, birth rate, and mortality rate.