Trial by fire: My mission to prove every real analysis theorem from scratch

Proving stuff yourself is hard, and my initial attempts were catastrophic. I wrote one proof that began with a stronger statement than the conclusion I was attempting to draw, rendering the proof useless even in the unlikely case it was otherwise successful. It was not otherwise successful. Jumping through several dodgy implications and logical errors, it eventually arrived at a conclusion that was substantially weaker than what I was attempting to prove. The worst part? I couldn’t even spot my own mistakes, and I thought the proof was probably quite good.

This post is about my approach to real analysis. In many areas of theoretical research, proving things is of paramount importance. Sadly, I didn’t have the foresight to take a heavy math curriculum as an undergrad, and like many, my skills were lackluster at best. My solution? Prove every theorem in an intermediate analysis course.

Though my early attempts still make me shudder, and I’m too embarrassed to include them here, I learned from my mistakes. The process was simple. After attempting a proof I would seek feedback (often from my generous friend who was always happy to chat about math over a beer, thanks Luke!). Once I understood the errors, I would make flashcards. Rehearsing these flashcards meant I could avoid making the same mistake twice.

This approach to learning analysis is frustrating, but it gets results. As I went through, slowly at first, I gathered momentum. I made less mistakes, moved through the course faster, and began to ‘discover’ some nice ways to prove things.

One of the first proofs that I’m proud of was an attempt to show that if a series converges absolutely then it converges.

I like it because it is quite different to the lecture proof, as far as I know perfectly valid, and it took only a few lines to spell out. I also found a neat way to use a lemma we’d proved earlier:

Overall, it’s been exciting to build up my proving skills, and while this approach isn’t for everyone (mostly because it’s too annoying), I hope my journey is encouraging to anyone thinking about giving some higher level maths a go.