IOM 2016 Problem 2

Let $a_1, . . . , a_n$ be positive integers satisfying the inequality $\sum_{i=1}^{n}\frac{1}{a_n}\le \frac{1}{2}$ . Every year, the government of Optimistica publishes its Annual Report with n economic indicators. For each $i = 1, . . . , n$ ,the possible values of the $i-th$ indicator are $1, 2, . . . , a_i$ . The Annual Report is said to be optimistic if at least $n - 1$ indicators have higher values than in the previous report. Prove that the government can publish optimistic Annual Reports in an infinitely long sequence.