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I found a problem that $(a_n)_{n\in\mathbb{N}}$ is any sequence. If series series $|a_n- a_{n+1}|$ is convergent than it guarantees that sequence $(a_n)_{n\in\mathbb{N}}$ is convergent?? I am unable to approach the problem since I want to disprove it with counter example which I am not getting ...... Can anybody help me solve it.

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Since $\sum_n |a_n-a_{n+1}|$ converges, so does $\sum_n (a_n-a_{n+1})$.

The latter series telescopes, yielding convergence of $a_n$.

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