Question 90
Prove that for any real number $M \gt 2$, there exists a strictly increasing infinite sequence of positive integers $a_1, a_2, \dots$ satisfying both the following two conditions:
(1) $a_i \gt M^i$ for any positive integer $i$.
(2) An integer $n$ is non-zero if and only if there exists a positive integer $m$ and $b_1, b_2, \dots, b_m \in \{-1, 1\}$, with $n = b_1a_1 + b_2a_2 + \dots + b_m a_m$.
(1) $a_i \gt M^i$ for any positive integer $i$.
(2) An integer $n$ is non-zero if and only if there exists a positive integer $m$ and $b_1, b_2, \dots, b_m \in \{-1, 1\}$, with $n = b_1a_1 + b_2a_2 + \dots + b_m a_m$.