Classification of the Computable Approximations by Divergence Boundings 1

A real number is called computably approximable if there is a computable sequence of rational numbers which converges to it. To investigate the complexity of computably approximable real numbers, we can consider the converging speed of the sequences. In this paper we introduce a natural way to measure the converging speed by counting the jumps of certain size appeared after certain stages. The number of this big jumps can be bounded by a bounding function. For different choice of bounding functions, we introduce various classes of real numbers with different approximation quality and discuss their mathematical properties as well as computability theoretical properties.