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Definition: a directed set is a set $M$ together with a preorder $\geq$ (reflexive and transitive order) such that every pair of elements in $M$ has an upper bound ($\forall x,y \in M, \ \exists z \in M,\ z \geq x \wedge z \geq y$)

For sequences in the reals, there always exists a monotone subsequence. The proof of this fact uses the total order property of the real numbers. But I was wondering, would it work for directed sets? In other words, does a sequence with values in a directed set $(M, \geq)$ always have a monotone subsequence?

I tried to find a counterexample, but I have yet to work with directed sets to be able to visualize them at this level.

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Let $M=\wp(\Bbb N)$, directed by inclusion, and let $a_n=\{n\}$ for $n\in\Bbb N$; then $\langle a_n:n\in\Bbb N\rangle$ is a sequence in $M$ with no monotone subsequence.

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  • $\begingroup$ Spot on! Thank you! $\endgroup$ – Stefan Octavian Jan 22 at 21:02
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    $\begingroup$ @StefanOctavian: You’re welcome! $\endgroup$ – Brian M. Scott Jan 22 at 21:03
  • $\begingroup$ @BrianM.Scott Hi professor Scott, could I ask your assistance here, please? I have to prove that two set are disjoint but unfortunately it seems I do not be able to prove it. Excuse me for the bother. $\endgroup$ – Antonio Maria Di Mauro Jan 22 at 22:50
  • $\begingroup$ @AntonioMariaDiMauro: I’m afraid that you’re now getting into an area about which I know virtually nothing. $\endgroup$ – Brian M. Scott Jan 23 at 0:24

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