Based on the notion of the E-subgradient, we present a unified tech nique to establish convergence properties of several methods for nonsmooth convex minimization problems. Starting from the technical results, we obtain the global convergence of : (i) the variable metric proximal methods presented by Bonnans, Gilbert, Lemarechal, and Sagastizabal, (ii) some algorithms proposed by Correa and Lemarechal, and (iii) the proximal point algorithm given by Rockafellar. In particular, we prove that the Rockafellar-Todd phenomenon does not occur for each of the above mentioned methods. Moreover, we explore the convergence rate of {parallel to x(k)parallel to}and {f(xk)} when {x(k)} is unbounded and {f(x(k))} is bounded for the nonsmooth minimization methods (i), (ii), and (iii).