ILL 1986 Problem 73

Let $(a_i)_{i\in \mathbb N}$ be a strictly increasing sequence of positive real numbers such that $\lim_{i \to \infty} a_i = +\infty$ and $a_{i+1}/a_i \leq 10$ for each $i$ . Prove that for every positive integer $k$ there are infinitely many pairs $(i, j)$ with $10^k \leq a_i/a_j \leq 10^{k+1}.$