There are two important classes of proofs not explicitly mentioned in other answers -

- Where the theorem is correct in all cases of interest, but not strictly correct as stated;
- Where the theorem is true but the proof fails to prove it.

These are distinct from when the theorem is Just Plain Wrong, something which you could tell more easily. I think professional mathematicians are more likely to produce a near-correct argument than a proof of something outright false, compared to people working at a more elementary level.

By the first of these classes, I mean that there may be missteps in how the theorem is stated, such that it holds for the "important" examples, but may fail in edge cases. For example, maybe it is stated for all sets $S$, but actually only holds if $S$ is nonempty. Sharpening up the theorem statement can help to bring out more subtle presentational issues as well. I would tend to go through the conditions used in the theorem and check against extremal cases like "but what if $n=0$?", "does $A$ really have to be finite?", that kind of thing.

This issue can occur more seriously in disguise. I recall a paper (not by me) which proved that all mappings with a certain property had a unique fixed point. A critical step was a lemma that derived a contradiction from the existence of two distinct fixed points, hence concluding uniqueness. But the authors had not noticed that some of their mappings had no fixed points at all. What they had really proved was that there was *at most one* fixed point. (They were generalizing from a previous result where there always was a fixed point, but their new version of the key property did not guarantee that.) The problem arose from the way that they were focused on the "good" cases, for which the proof worked. They might have noticed it if they had been more explicit about naming the set $S$ of fixed points, and therefore getting a "hook" into considering the various cases of $S$ being empty, a singleton, etc.

My second class of proofs is more tricky, because the theorem is actually correct. It's just that the argument for it is flawed. Often, the proof *does* work for a subset of the scenarios envisaged in the theorem statement, but fails to go through more generally. So it is allied to my first class of examples. For example, I think it's fairly common for well-ordering or choice issues to sneak their way into the middle of a proof, *and* for the use of choice to be ultimately unnecessary.

What we are looking for to detect these problems is not a counterexample to the theorem (there isn't one), but a counterexample to the proof. So again, it helps to run through the proof steps "adversarially", trying to pick the most extreme or awkward cases. When you read something like "let $x$ be the minimal such doodad", focus on that "the" and try to break it with an example where there are zero or many possibilities.

I've also found it helpful to restructure big proofs, so that there are more and smaller lemmas leading up to the main result. While this does add length, it has the benefits of:

- Shorter lemmas are easier to check in isolation.
- Making the intermediate structures clearer is often helpful - even just by giving names to frequently-used concepts.
- Having to formalize the lemma statement forces clarity over what it is claiming, vs. having it embedded seamlessly in a longer context.

Another little trick for incorrect proofs of $A \implies B$ is to try to recast them as proofs of $\neg B \implies \neg A$. By focusing on $\neg B$, you can avoid the trap of accidentally relying on the most favourable cases of $A$.

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