IMO 1988 LL FRA9

If a0 is a positive real number, consider the sequence {an}

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IMO 1988 LL FRA9

Origin: FRA

Problem

If a0 is a positive real number, consider the sequence {an} defined by an+1 = a2 n −1 n + 1 for n \geq0. Show that there exists a real number a > 0 such that: (i) for all real a0 \geqa, the sequence {an} \to+\infty(n \to\infty); (ii) for all real a0 < a, the sequence {an} \to0.