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 (x(n))(nepsilonN) of rational numbers which converges to x such that the convergence is k-monotonic in the sense that k . \x - x(n) \ greater than or equal to \x - x(m)\ for any m > n and x is monotonically computable (me) if it is k-mc for some k > 0. x is weakly computable if there is a computable sequence (x(s))(sepsilonN) of rational numbers converging to x such that the sum Sigma(sepsilonN) \x(s) - x(s+1)\ is finite. In this paper we show that all me real numbers are weakly computable but the converse fails. Furthermore, we show also an infinite hierarchy of me real numbers.