The gradient sampling method is a recently developed tool for solving unconstrained nonsmooth optimization problems. Using just first-order information about the objective function, it generalizes the steepest descent method, one of the most classical methods for minimizing a smooth function. This study aims at determining under which circumstances one can expect the same local convergence result of the Cauchy method for the gradient sampling algorithm under the assumption that the problem is stated by a finite max-function around the optimal point. Additionally, at the end, we show how to practically accomplish the required hypotheses during the execution of the algorithm.