Question 6
The sequence $a_1,a_2,\dots$ of integers satisfies the conditions:
(i) $1\le a_j\le2015$ for all $j\ge1$,
(ii) $k+a_k\neq \ell+a_\ell$ for all $1\le k\lt \ell$.
Prove that there exist two positive integers $b$ and $N$ for which $$
\left\lvert\sum_{j=m+1}^n (a_j-b) \right\rvert \le 1007^2
$$ for all integers $m$ and $n$ such that $n \gt m\ge N$.
(i) $1\le a_j\le2015$ for all $j\ge1$,
(ii) $k+a_k\neq \ell+a_\ell$ for all $1\le k\lt \ell$.
Prove that there exist two positive integers $b$ and $N$ for which $$
\left\lvert\sum_{j=m+1}^n (a_j-b) \right\rvert \le 1007^2
$$ for all integers $m$ and $n$ such that $n \gt m\ge N$.