The problem (taken from Kaplansky's Set theory and metric spaces, p. 40) is as follows
Let $L$ be a lattice in which every chain has a least upper bound and a greatest lower bound. Prove that $L$ is complete.
Using the fact that $L$ is a lattice in which every chain has an upper bound I was able to prove that $L$ has a unique maximal element (with Zorn's lemma). With this I can prove that an empty set and any subset of $L$ containing its maximal element have a least upper bound.
But I get stuck trying to prove that a non-empty subset of $L$ not containing the maximal element has a least upper bound. How can I do this? Note that I can not use the axiom of choice here, only Zorn's Lemma.