CGMO 2003 Problem 5

Let $\{a_n\}^{\infty}_1$ be a sequence of real numbers such that $a_1 = 2,$ and $$a_{n+1} = a^2_n - a_n + 1, \forall n \in \mathbb{N}.$$ Prove that $$1 - \frac{1}{2003^{2003}} < \sum^{2003}_{i=1} \frac{1}{a_i} < 1.$$