MathDB

Consider a sequence of numbers between

0

and

1

in which the next number after

x

1 - |1 - 2x|

. (

|x| = x

x \ge 0

|x| = -x

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)

Problem Statement