Being too pedantic with writing proofs

Solution 1:

I've looked at your three proofs, but I only analyzed the first one very closely (since the answerers of the last two provided detailed comments on your proofs). I've added an answer to the first question.

You'll notice that in my answer I use a very common lemma: $B \subseteq \cap_{i \in I} A_i$ iff $\forall i \in I \ B \subseteq A_i.$

Everyone learns these kinds of lemmas eventually, usually from reading proofs that use them. Sometimes one discovers them on one's own, but this usually ultimately depends on inspiration from encountering broadly similar arguments in other people's proofs first.

I'm not very familiar with Velleman's book, but from looking at some of it casually, it seems that most of the arguments presented go back to the level of elements rather than using any kind of higher-level lemmas on sets. So you can hardly be blamed for reproducing the same style of proof the author uses.

Your proofs will naturally become more sophisticated when you start reading more sophisticated mathematics. In the meantime, you're doing the right thing by breaking things down so that you understand every detail of a proof. That's the main thing.

Another way you can improve your proofs is by selecting textbooks or problem books with full solutions. That way you can compare your solution with the book's. You seem to have the discipline to do things on your own before looking at a solution, so this is likely to be a help to you, not a hindrance.

Solution 2:

Here are my two cents. I've only read the second proof you linked to, and only the first direction of it, and I think it is not too pedantic at all.

The thing it, it all depends on how far gone you are in your mathematical studies and how used to you are, and how comfortable you're with, rigorous proofs. Since you are just beginning studying rigorous mathematics - and self-studying at that - I think it is actually crucial you begin by writing such "pedantic" proofs. Then, once you grow familiar with such proofs and become more confident, you can start writing more 'casual' proofs, because you'd develop an instinct that the proof is really correct and that you could make it completely rigorous if you needed to.

This doesn't stop at your level: the more you study, the less rigorous proofs becomes. And in fact, people often make mistakes, thinking something is obvious and that they could prove it completely rigorously if they wanted to, which then later turns out to be false. But that's just the way these things work, and you have to at least become a little more relaxed at proof-writing, or else you'd never have the time to prove anything more "serious", involving more complex mathematics.

One thing which is perhaps confusing about this is the feeling that you can always be more pedantic. And it's more or less true. I suggest reading about formal languages and formal proofs, if you find this point interesting/confusing. But in my opinion (and most people's) the level of rigor in the proofs you linked to (at least what I read) is enough. Why? Because, usually, when you write at such a level of rigor, even in the beginning of your studies, you don't make mistakes, thinking something is clear while it is in fact incorrect. That is, since your proofs are fairly rigorous (even if they could be even more rigorous), there aren't really too many "subtle" points you might be missing.

This is at least how I see it.