Does any unbounded directed set contain an unbounded, increasing sequence?

Question

Assume $D$ is a directed set such that there does not exist $d\in D$ satisfying $d\geq d'$ for all $d'\in D$.

Can we always find a sequence $(d_n)_{n\in \mathbb N}$ such that $d_{n+1}\geq d_n$ for all $n$, and such that there does not exist $d\in D$ satisfying $d\geq d_n$ for all $n\in \mathbb N$?

Is the result true? Is there an easy counter-example? Or only under some assumptions?

Answer 1

Let $D$ be the first uncountable ordinal.