Let be a Banach space and a contraction mapping, where is a closed set. Consider a sequence and define the sequence , by , where is a sequence of natural numbers. We highlight some general conditions so that the two sequences and are simultaneously convergent. Both cases: 1) , for all , and 2) , for all , are discussed. In the first case, a general Picard iteration procedure is inferred. The results are then extended to
sequences of mappings and some appropriate applications are also proposed.