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Part of 1989 Nordic

Problems(1)

x_0 = t, x_{n+1} = 2x_n^2-1, x_n = 1

Source: Nordic Mathematical Contest 1989 #3

10/5/2017
Let SS be the set of all points tt in the closed interval [−1,1][-1, 1] such that for the sequence x0,x1,x2,...x_0, x_1, x_2, ... defined by the equations x0=t,xn+1=2xn2−1x_0 = t, x_{n+1} = 2x_n^2-1, there exists a positive integer NN such that xn=1x_n = 1 for all n≥Nn \ge N. Show that the set SS has infinitely many elements.
elements of setSequencealgebra