Difficult undergrad proofs

Dear all:

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 X 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.

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9 Responses to Difficult undergrad proofs

  1. Adam Azzam says:

    I agreed with Munkres that Urysohn’s (Lemma and) Metrization Theorem was the first deep theorem of topology that I’d learned. I’ll think of some more and leave them in a comment.

    • Z Norwood says:

      That’s probably a good example of what I had in mind. I don’t remember the proof of Urysohn, but it’s probably at least 60% technical. I don’t remember the proof being particularly informative, either; the result is what’s interesting.

  2. JCummings says:

    Do proofs from probabilistic combi count? I was actually just telling Adam yesterday about a proof by Erdős on sum-free sets that is amazingly clever. There is no way I would have ever thought of it. If I at some point I write on the probabilistic method it will definitely be included.

    Also, I believe it was more of how \mu^*-measurability was defined, but I am reminded about one measure theory lecture which Dr. Radcliffe opened as: “Today we are going to start proving Carathéodory’s theorem today. It relies on an idea that is …. really fucking brilliant.”

    • Z Norwood says:

      I like your probabilistic combi example (and I think I remember you getting worked up about it!).

      The example from Measure Theory isn’t exactly what I had in mind, though, since that trick was just meant to streamline the proof; it wasn’t essential. The slick proof might not have been the proof we would have come up with, but I think any one of us could have (eventually) proved that theorem without having seen the proof (especially with a hint).

  3. Ryan says:

    Mine have all been said so far =( Jay took mine about the idea behind Caratheodory’s definition of measurable sets, and I was also going to mention Urysohn’s lemma. Along that line, though we didn’t actually prove it in undergrad, we’ve worked through this in my topology class this year, the proof of Tychonoff’s Theorem I have seen is fairly clever. It uses nets, which are a fairly intuitive generalization of sequences, but then it also introduces the idea of a universal net, which is really wacky and something I would never think of on my own.

    • Z Norwood says:

      Tychonoff is a good example. Have you seen the proof of Tychonoff using ultrafilters? It’s nifty. Ultrafilters are always the way to go!

  4. JCummings says:

    Do any of you remember that long, difficult (I don’t remember if it was technical-difficult or clever-difficult) proof that Pitts had us read in 496? I forget what it was about, but I think it was the most difficult, tricky, and/or annoying proof from that class.

    It don’t believe it was in Axler or Halmos. I remember reading it from a print-out.

    • Z Norwood says:

      No, but since it was from Pitts’ class, it almost certainly would have been the one I would have been least likely to prove on my own.

  5. Z Norwood says:

    No examples from 850/852, anyone? Surely there was at least one proof in there that required at least two tricks.

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