The other day I was reflecting on what material we covered in our undergrad courses, and I wondered what the most difficult proof we encountered in courses there was. Of course, we should keep in mind that every proof seems considerably less insightful (often obvious) once one understands it, but there are definitely some proofs that require deeper insight than others.
My question to you: What proof from our undergrad courses was the most difficult? Were there any that can’t reasonably be summaris(z)ed by a sentence of the form “Perform [possibly clever] trick and then do the obvious things”? I think a lot of the proofs in combi/graph theory required some sort of clever technique, but I can’t think of one that definitely required more than one bit of cleverness (though I don’t remember them all, of course).
Clarification: Obviously there were plenty of technical proofs. That’s not what I mean. For example, the proof (due to Blass) I described in my most recent post on AC that uniqueness of dimension implies AC requires more than just one simple insight, I think.
If you don’t like my question in any of the forms it’s appeared in so far, how about this one: Which proof would you have been least likely to come up with yourself, if you were asked to prove a theorem (possibly with one one-line hint)?
Also, Happy Thanksgiving.