Weakly computable real numbers

A real number x is recursively approximable if it is a limit of a computable sequence of rational numbers. If, moreover. the sequence is increasing (decreasing or simply monotonic), then x is called left computable (right computable or semi computable). x is called weakly computable if it is a difference of two left computable real numbers. We show that a real number is weakly computable if and only if there isa computable sequence (x(s))(s is an element ofN) of rational numbers which converges to x weakly effectively, namely the sum of jumps of the sequence is bounded. It is also shown that the class of weakly computable real numbers extends properly the class of semi-computable real numbers and the class of recursively approximable real numbers extends properly the class of weakly computable real numbers. (C) 2000 Academic Press.