We are wanting to show that. Given a sequence $(a_n)$ such that the series $\sum_{n=1}^\infty|a_{n+1} - a_n|$ converges, show that $(a_n)$ converges.

My idea was to use the partial $m$-th sum of this series, then split it up into the $S_{2m}$ and $S_{2m+1}$ sums of $(a_n)$ and show that both of them converge. However, I am having trouble with this as we do not know if the sequences are monotonic or not. Would this be the right way to go about doing it?

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    $\begingroup$ Please visit this page to learn how to typeset your question in MathJax. This will help draw interest and make it more readable. Someone has already edited most of your post; I might suggest learning the MathJax presented here for future questions. $\endgroup$ – Clayton Aug 29 '18 at 0:51

Since $\sum|a_{k+1} - a_k|$ is convergent, the Cauchy criterion is satisfied by partial sums. For all $\epsilon > 0$ there exists $N \in \mathbb{N}$ such that if $m > n > N$ we have

$$|a_m - a_n| \leqslant \sum_{k=n}^{m-1} |a_{k+1} - a_k| = \left|\sum_{k=1}^{m-1} |a_{k+1} - a_k| - \sum_{k=1}^{n-1} |a_{k+1} - a_k| \right|< \epsilon$$

Therefore, $(a_n)$ is a Cauchy sequence and, hence, convergent.

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  • $\begingroup$ I do not fully grasp where that initial inequality comes from. How do we know that this inequality holds? $\endgroup$ – Matt_G Aug 29 '18 at 1:13
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    $\begingroup$ The first one is a repeated application of the triangle inequality, eg. $|a_3 - a_2 + a_2 - a_1| \leqslant |a_3-a_2| + |a_2-a_1|$ $\endgroup$ – RRL Aug 29 '18 at 1:20

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