MathDB

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}.

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