NO-ABEL 1998 Problem 1

Let $a_0,a_1,a_2,...$ be an infinite sequence of positive integers such that $a_0 = 1$ and $a_i^2 > a_{i-1}a_{i+1}$ for all $i > 0$ . (a) Prove that $a_i < a_1^i$ for all $i > 1$ . (b) Prove that $a_i > i$ for all $i$ .