A sequence \(x_1, x_2, x_3, \dots\) has the following properties:

(a) \(1=x_1 < x_2 < x_3 \dots\);

(b) \(x_{n+1} \leq 2n\) for all \(n \in \mathbb{N}\).

Prove that for each positive integer \(k\) there exist indices \(i\) and \(j\) such that \(k = x_i −x_j\).

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