[eng] George and Veeramani characterized complete fuzzy metric spaces (X;M; ) by means of nested sequences fAng of closed sets of X which have fuzzy diameter zero [3]. In [5] an appropriate concept of p-Cauchy sequence, according to the concept of p-convergence due to D. Mihet [11], was given. In this paper we introduce for fAng a concept of p-fuzzy diameter zero, which is according to the concept of p-convergence. Then, we characterize by means of certain nested sequences fAng which have p-fuzzy diameter zero, those fuzzy metric spaces in which p-Cauchy sequences are convergent (p-convergent), called p- complete spaces (w-p-complete spaces). As a consequence of our results we obtain the well-known characterization of a complete