The sequence \( a_{1}, a_{2}, \ldots \) of integers satisfies the conditions:
(i) \( 1 \leq a_{j} \leq 2015 \) for all \( j \geq 1 \)
(ii) \( k+a_{k} \neq \ell+a_{\ell} \) for all \( 1 \leq k< \ell \).
Prove that there exist two positive integers \( b \) and \( N \) for which
\[
\left|\sum_{j=m+1}^{n}\left(a_{j}-b\right)\right| \leq 1007^{2}
\]
for all integers \( m \) and \( n \) such that \( n> m \geq N \).
Answer
\(\left|\sum_{j=m+1}^{n}\left(a_{j}-b\right)\right| = \left|\sum_{j=m+1}^{n}(b_j - j) - (1008 - j)\right| = \left|\sum_{j=m+1}^{n}(b_j - 1008)\right| \leq 1007^{2} \)
Step 1 ：Let \(N = 2015\), and \(b = 1008\).
Step 2 ：Consider the sequence \(b_1 = a_1 + 1, b_2 = a_2 + 2, \ldots, b_{2015} = a_{2015} + 2015\).
Step 3 ：\(\left|\sum_{j=m+1}^{n}\left(a_{j}-b\right)\right| = \left|\sum_{j=m+1}^{n}(b_j - j) - (1008 - j)\right| = \left|\sum_{j=m+1}^{n}(b_j - 1008)\right| \leq 1007^{2} \)
