# Is a sequence convergent if it is bounded and a combination of a monotone increasing and decreasing sequence

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?

• 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. Sep 22, 2020 at 16:30

No. Check these $$x_n = (-1)^n, y_n = 2^n +(-1)^n, z_n = -2^n$$.
• $x_n$ is bounded, and $y_n, z_n$ need not be bounded in given assumption, just monotonicities.