Question 148
Let $\{a_n\}_1^\infty$ be a sequence of real numbers such that $a_1 = 2$, $a_{n+1} = a_n^2 - a_n + 1$, for $n = 1, 2, \dots$. Prove that
$$
1 - \frac{1}{2^{003^{2^{003}}}} \lt \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_{2^{003}}} \lt 1.
$$
(posed by Li Shenghong)
$$
1 - \frac{1}{2^{003^{2^{003}}}} \lt \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_{2^{003}}} \lt 1.
$$
(posed by Li Shenghong)