Consider a sequence of numbers between 0 and 1 in which the next number after x is 1−∣1−2x∣. (∣x∣=x ifx≥0, ∣x∣=−x if x<0.) Prove that(a) if the first number of the sequence is rational, then the sequence will be periodic (i.e. the terms repeat with a certain cycle length after a certain term in the sequence);
(b) if the sequence is periodic, then the first number is rational. (G Shabat) algebrarationalSequenceperiodic