State and prove Zorn's Lemma. [You may assume Hartogs' Lemma.] Indicate clearly where in your proof you have made use of the Axiom of Choice.

Show that $\mathbb{R}$ has a basis as a vector space over $\mathbb{Q}$.

Let $V$ be a vector space over $\mathbb{Q}$. Show that all bases of $V$ have the same cardinality.

[Hint: How does the cardinality of $V$ relate to the cardinality of a given basis?]