In my problem, I have a specific poset which is also a lattice, but I think the result holds for any finite poset, so let $(P,\preceq)$ be a finite poset with $\#P=n$ and maximal element $M$ and minimal element $m$. I want to show that I can write $P=\{p_1,...,p_n\}$ such that for any $i,j$ with $i<j$, we don't have $p_i \succ p_j$ or equivalently, either $p_i \preceq p_j$ or $p_i, p_j$ are incomparable. Obivously we need $p_1=m$ and $p_n=M$, but other than that I'm pretty stuck.
My idea was to take some $p \in P\cap \{m,M\}^c$. Then, if $p$ is incomparable to any other element excepts $m,M$, then put $p_2=p$. Otherwise, $p$ might be comparable to elements say $q_1,\dots q_k$. If all these are larger than $p$, again we can put $p_2=p$. Otherwise, say $q_1,\dots, q_\ell \prec p_2$ and hence all these $q$'s must come before $p$ in the list.
Then you would need to go through the same kind of arguments for each of the $q_1,\dots,q_\ell$, and this is where I get lost. I assume at some point we must use that $P$ is finite such that this process doesn't continue forever...