(i) State and prove Zorn's Lemma. [You may assume Hartogs' Lemma.] Where in your proof have you made use of the Axiom of Choice?

(ii) Let $<$ be a partial ordering on a set $X$. Prove carefully that $<$ may be extended to a total ordering of $X$.

What does it mean to say that $<$ is well-founded?

If $<$ has an extension that is a well-ordering, must $<$ be well-founded? If $<$ is well-founded, must every total ordering extending it be a well-ordering? Justify your answers.