Dear Saddened Exam Students,
My hints will have to be kept to a minimum because (i) this is an exam after all, and (ii) not every student will necessarily look at the forum and I want to be fair. However, it you ask some more pointed questions, perhaps I can provide nudges in the correct direction.
I want to emphasize that you have all the tools you need to tackle these problems. Yet the days of mimicking examples are over. Rarely, can you apply one proof to another theorem with only slight modifications. An example of when this does work is the proof of $\sqrt{2}$ being irrational can be tweaked slightly to prove that $\sqrt{5}$ is irrational. Writing proofs usually requires you to use your own awesome prodigious mental faculties. This is one of the appealing aspects of proof-writing and probably one of the top reasons mathematicians become mathematicians. I also realize that writing proofs is outside of your comfort zone, which is exactly why we are asking you to tackle them in small doses in this course. Moreover, I want to emphasize that you shouldn't expect (although it might happen) to figure out a proof in 10 minutes. I've worked on proving a few theorems for years and still not figured them out. I've also spent years working on proving a big theorem and figured it out. It is an amazing feeling when you get it and totally humbling when you don't. It wouldn't surprise me (and I mostly expect) to hear that some of you spent 3 solid hours or more (minus anytime you spent complaining about how hard it was:) ) working on just one proof. Welcome to mathematics! Buckle up and enjoy the ride.
