On a notion of rapidity of convergence used in the study of fixed point iterative methods

There exist many papers written in the last 50 years or so which are using a concept of rapidity of convergence for two comparable sequences. It appears that the original source of this notion is not known to most of the authors of those papers, since for this notion they are refereeing to different sources or simply do not refer to any other publication like this notion would be ”folklore” or even like they would be the ones who introduced for the first time such a notion. Starting from this fact, the aim of the present note is to try to find out who was the true author of that notion of rapidity of convergence and, secondly, to illustrate how it is used in the particular case of the study of fixed point iterative methods.