Problem 1102 (difficulty: 8/10)

Let \(\displaystyle \sum\limits_{n=1}^n a_n\) be a divergent series with positive terms. Prove that there is a sequence \(\displaystyle c_n\) of positive numbers, such that \(\displaystyle c_n \to 0\) as \(\displaystyle n \to \infty\) and \(\displaystyle \sum\limits_{n=1}^n (c_n\cdot a_n)\) still diverges.

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