ILL 1988 Problem 9

If $a_0$ is a positive real number, consider the sequence $\{a_n\}$ defined by:

$$a_{n+1} = \frac{a^2_n - 1}{n+1}, n \geq 0.$$

Show that there exist a real number $a > 0$ such that:

i.) for all $a_0 \geq a,$ the sequence $\{a_n\} \rightarrow \infty,$

ii.) for all $a_0 < a,$ the sequence $\{a_n\} \rightarrow 0.$