Weakly computable real numbers

A real number x is recursively approximable if it is a limit of a computabl e sequence of rational numbers. If, moreover. the sequence is increasing (d ecreasing or simply monotonic), then x is called left computable (right com putable or semi computable). x is called weakly computable if it is a diffe rence of two left computable real numbers. We show that a real number is we akly 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 t he class of weakly computable real numbers extends properly the class of se mi-computable real numbers and the class of recursively approximable real n umbers extends properly the class of weakly computable real numbers. (C) 20 00 Academic Press.