The convergence of a general algorithm for limit analysis involving rate‐sensitive materials is proved. The application of this combined smoothing and successive approximation (CSSA) algorithm was extended to deal with plasticity problems involving rate‐sensitive materials. A plasticity problem was stated by the upper bound formulation derived rigorously from the lower bound formulation. Applying a finite‐element discretization, we then employed the CSSA algorithm to solve the resulting optimization problem iteratively. Finally, the Holder inequality was adopted to prove the convergence of the CSSA algorithm. Moreover, it is the familiar Cauchy‐Schwarz inequality, a reduced form of the Hölder inequality, which is utilized to prove the convergence of the CSSA algorithm applied to plasticity problems involving rate‐insensitive materials.