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.
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.