International Mathematical Olympiad 2015 Problem 6

The sequence $a_1, a_2, \ldots$ of integers satisfies the following conditions:

(i) $1 \leqslant a_{j} \leqslant 2015$ for all $j \geqslant 1$;

(ii) $k + a_{k} \neq \ell + a_{\ell}$ for all $1 \leqslant k < \ell$.

Prove that there exist two positive integers $b$ and $N$ such that $$\left| \sum_{j = m + 1}^{n} (a_{j} - b) \right| \leqslant 1007^{2}$$ for all integers $m$ and $n$ satisfying $n > m \geqslant N$.