TUYMAADA 2005 Problem 6

Given are a positive integer $n$ and an infinite sequence of proper fractions $x_0 = \frac{a_0}{n}$ , $\ldots$ , $x_i=\frac{a_i}{n+i}$ , with $a_i < n+i$ . Prove that there exist a positive integer $k$ and integers $c_1$ , $\ldots$ , $c_k$ such that $$c_1 x_1 + \ldots + c_k x_k = 1.$$

Proposed by M. Dubashinsky