ISL 1979 Problem 53

An infinite increasing sequence of positive integers $n_j (j = 1, 2, \ldots )$ has the property that for a certain $c$ , $$\frac{1}{N}\sum_{n_j\le N} n_j \le c,$$ for every $N >0$ . Prove that there exist finitely many sequences $m^{(i)}_j (i = 1, 2,\ldots, k)$ such that $$\{n_1, n_2, \ldots \} =\bigcup_{i=1}^k\{m^{(i)}_1 ,m^{(i)}_2 ,\ldots\}$$ and $$m^{(i)}_{j+1} > 2m^{(i)}_j (1 \le i \le k, j = 1, 2,\ldots).$$