If a lattice is countable, prove that it has a subset that is both totally ordered and cofinal in the lattice. Cofinal means that for each $l$ in the lattice, there is some $a$ in the subset such that $l\le a$.
My idea was to try to use Zorn's lemma on the set of all totally ordered subsets and prove it has a maximal element which must be cofinal, but this hasn't helped much.
First list the elements of the lattice in a sequence (possible as the lattice is countable). Then define a new sequence whose $n$-th term is the join of the first $n$ terms of your original sequence. This new sequence is totally ordered and cofinal in the lattice.