Question 11
Consider an infinite sequence $a_{1}, a_{2}, \ldots$ of positive integers with $a_{i} \leqslant 2015$ for all $i \geqslant 1$. Suppose that for any two distinct indices $i$ and $j$ we have $i+a_{i} \neq j+a_{j}$.
Prove that there exist two positive integers $b$ and $N$ such that
$$
\left|\sum_{i=m+1}^{n}\left(a_{i}-b\right)\right| \leqslant 1007^{2}
$$
whenever $n\gt m \geqslant N$.
Prove that there exist two positive integers $b$ and $N$ such that
$$
\left|\sum_{i=m+1}^{n}\left(a_{i}-b\right)\right| \leqslant 1007^{2}
$$
whenever $n\gt m \geqslant N$.