MathDB

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).

Problem Statement