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I try to prove this by proving $(a_n)$ is a Cauchy sequence. First, since $s_n = \sum|a_n - a_{n+1}| < \infty$, $s_n$ converges and thus $$\lim_{n\rightarrow \infty} |a_n - a_{n+1}| = 0$$ Then, for every $\epsilon > 0$, let $n_0 \in \mathbb{N}$ such that $\forall n \geq n_0, |a_n - a_{n+1}| < \frac{\epsilon}{n_0}$. Then I have $$|a_n - a_m| \leq |a_n - a_{n+1}| + \cdots+|a_m-1 + a_m| \\ < (m-n-1)\frac{\epsilon}{n_0}$$ Here I got stuck, what if $\frac{m-n-1}{n_0}>1$? How can I rewrite this equation so that it will be exactly less than $\epsilon$?Can anyone give me some suggestions?