Problem

Source: 2018 China Southeast MO Grade 11 P2

Tags: algebra



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