(i) State Zorn's Lemma. Use Zorn's Lemma to prove that every real vector space has a basis.

(ii) State the Bourbaki-Witt Theorem, and use it to prove Zorn's Lemma, making clear where in the argument you appeal to the Axiom of Choice.

Conversely, deduce the Bourbaki-Witt Theorem from Zorn's Lemma.

If $X$ is a non-empty poset in which every chain has an upper bound, must $X$ be chain-complete?