Monotonically Computable Real Numbers

A real number x is called k-monotonically computable (k-mc), for constant k > 0, if there is a computable sequence (xn)n∈N of rational numbers which converges to x such that the convergence is k-monotonic in the sense that k · |x−xn| ≥ |x−xm| for any m > n and x is monotonically computable (mc) if it is k-mc for some k > 0. x is weakly computable if there is a computable sequence (xs)s∈N of rational numbers converging to x such that the sum ∑ s∈N |xs− xs+1| is finite. In this paper we show that all mc real numbers are weakly computable but the converse fails. Furthermore, we show also an infinite hierarchy of mc real numbers.