A poset, all of whose nonempty subsets have a maximum and a minimum, is finite

Question

Let $X$ be a poset such that all nonempty subsets of $X$ have a maximum and a minimum.

Show that $X$ is finite.

Answer 1

HINT: Since you’ve already shown that $X$ is linearly ordered, you can argue as follows. Let $x_0=\min X$. If you’ve already defined $x_k$ for $k<n$, and $X\ne\{x_k:k<n\}$, let $x_n=\min(X\setminus\{x_k:k<n\})$. If at some point $X=\{x_k:k<n\}$, you’re done. Otherwise ... ?