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If a bounded sequence is the sum of a monotone increasing and a monotone decreasing sequence ($x_n=y_n+z_n$ where ${y_n}$ is monotone increasing and ${z_n}$ is monotone decreasing) does it follow that the sequence converges?

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  • $\begingroup$ The difference of non-decreasing bounded sequences is a sequence of bounded variation. The converse is also valid. That's the best result in this direction that I know of. $\endgroup$ Sep 22, 2020 at 16:30

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No. Check these $x_n = (-1)^n, y_n = 2^n +(-1)^n, z_n = -2^n $.

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    $\begingroup$ $x_n$ is bounded, and $y_n, z_n$ need not be bounded in given assumption, just monotonicities. $\endgroup$
    – Han
    Sep 22, 2020 at 16:15
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    $\begingroup$ You're right, my mistake. $\endgroup$
    – Integrand
    Sep 22, 2020 at 16:29

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