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Find all initial values which make the sequence bounded.

Source: 2018 China Southeast MO Grade 11 P2

July 30, 2018
algebra

Problem Statement

Suppose that aa is real number. Sequence a1,a2,a3,....a_1,a_2,a_3,.... satisfies a1=a,an+1={an−1an,an≠00,an=0(n=1,2,3,..)a_1=a, a_{n+1} = \begin{cases} a_n - \frac{1}{a_n}, & a_n\ne 0 \\ 0, & a_n=0 \end{cases} (n=1,2,3,..) Find all possible values of aa such that ∣an∣<1|a_n|<1 for all positive integer nn.
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