ILL 1988 Problem 80

The sequence $\{a_n\}$ of integers is defined by $$a_1 = 2, a_2 = 7$$

and $$- \frac {1}{2} < a_{n + 1} - \frac {a^2_n}{a_{n - 1}} \leq \frac {}{}, n \geq 2.$$

Prove that $a_n$ is odd for all $n > 1.$