Problem 225 (difficulty: 6/10)

Prove that if \(\displaystyle (a_n )\) is convergent and \(\displaystyle (a_{n+1} -a_n )\) is monotone, then \(\displaystyle n\cdot (a_{n+1} -a_n ) \to 0.\) Give an example for a convergent sequence \(\displaystyle (a_n )\) for which \(\displaystyle n\cdot (a_{n+1} -a_n ) \) does not tend to \(\displaystyle 0\).

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