"Strange" proofs of existence theorems

There are probabilistic proofs of existence. Do they fall into one of your three categories?

For example, prove the existence of a real number that is normal in all bases: To do it, we show that "almost all" real numbers (according to Lebesgue measure) have that property. Therefore at least one real number has the property. And the point is: this "almost all" proof is easier than constructing an explicit example.

See some nice examples due to Erdős in the cited Wikipedia page which use only finite probability spaces. If we show that a probability is ${} > 0$, then the set is not empty.

There is a famous proof of the existence of two irrational numbers $a$ and $b$ such that $a^b$ is rational. The proof considers two cases: $\sqrt{2}^\sqrt{2}$ is irrational, or it is rational. In either case we can find such $a$, $b$. Then it applies the law of excluded middle to say one of these cases in fact holds. You can see a discussion of the proof here: http://math.andrej.com/2009/12/28/constructive-gem-irrational-to-the-power-of-irrational-that-is-rational/.

You mentioned “proof by contradiction” in your question, but to me this application of the law of the excluded middle is conceptually different than proof by contradiction.

(By the way, as discussed in that blog post, this is certainly not a serious application of the law of excluded middle because there are other ways to prove the result in question. But it is a cute proof.)

EDIT: I believe there might be more serious proofs along these lines in number theory, that go something like “either the Riemann hypothesis holds, or it doesn’t. In the first case...; in the second case...” Or the same but with “Siegel zero existing.” But I don’t know a particular example off the top of my head.

Many existence proofs in analysis / probability follow this line of argument: 1. Construct a family of objects that approximately satisfy some desired property. 2. Show that the family is precompact. 3. Show that every accumulation point must satisfy the desired property. I suppose that to some extent this would often count as a constructive proof since in many cases one can impose additional constraints until the possible limits are reduced to a single point, but this may require some non-trivial amount of work...