Proofs for Undergraduates

Question

I am teaching a course in proof technique to undergraduate students. One of the things they can do for their project is read an involved proof and explain it, for example Gödel's Incompleteness Theorem. Does anyone have any good proofs? They can come from any field accessible to a student with a month to work on it and with knowledge of multivariable calculus and abstract algebra.

Answer 1

Would a proof of the compactness theorem for propositional logic be suitable as an "involved proof", not to be required of all students, but as an item on a menu from which each student may choose one?

Answer 2

In the same vein as Michael Hardy's proposal, how about the completeness theorem for the Hilbert calculus in propositional logic? It is fairly involved and introduces them to Zorn's lemma as well.