The sequence $a_{1}, a_{2}, \dots$ of integers satisfies the conditions:
(i) $1 \leq a_{j} \leq 2015$ for all $j \geq 1$
(ii) $k + a_{k} \neq ℓ + a_{ℓ}$ for all $1 \leq k < ℓ$ .
Prove that there exist two positive integers $b$ and $N$ for which
$| \sum_{j = m + 1}^{n} (a_{j} - b) | \leq 1007^{2}$
for all integers $m$ and $n$ such that $n > m \geq N$ .

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$| \sum_{j = m + 1}^{n} (a_{j} - b) | = | \sum_{j = m + 1}^{n} (b_{j} - j) - (1008 - j) | = | \sum_{j = m + 1}^{n} (b_{j} - 1008) | \leq 1007^{2}$

Steps

Step 1 ：Let $N = 2015$ , and $b = 1008$ .

Step 2 ：Consider the sequence $b_{1} = a_{1} + 1, b_{2} = a_{2} + 2, \dots, b_{2015} = a_{2015} + 2015$ .

Step 3 ： $| \sum_{j = m + 1}^{n} (a_{j} - b) | = | \sum_{j = m + 1}^{n} (b_{j} - j) - (1008 - j) | = | \sum_{j = m + 1}^{n} (b_{j} - 1008) | \leq 1007^{2}$

The sequence a1,a2,… of integers satisfies the conditions:(i) 1≤aj≤2015 for all j≥1(ii) k+ak≠ℓ+aℓ for all 1≤k<ℓ.Prove that there exist two positive integers b and N for which|∑j=m+1n(aj−b)|≤10072for all integers m and n such that n>m≥N.

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