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I know proof by example is one way, but I cannot remember what the other way is. I thought maybe it was proof by contradiction, but I cannot think of an example for the life of me.. so I'm not satisfied with assuming.

fractionofyoursmile
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    I also love to do math while drinking beer like now... but we need more details... – the_candyman Sep 28 '20 at 21:15
  • It's hard to say with certainty what you're thinking of, because there's no exhaustive list of how to prove things, but anything that can be proved at all can be proved by contradiction (even when that's not the best way). In the case of proofs of existence specifically, proof by contradiction is often contrasted with proof by example. (To wit, a proof by example is considered constructive or effective, while a proof by contradiction is not.) So if you think that you remember proof by contradiction, then that's probably what it was. – Toby Bartels Sep 28 '20 at 21:26

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Assume you want to prove the existence of a certain element $ c$. In general, there are two ways :

The first is using a Theorem which insure the existence of such element. For instance, the IVT says there exist an element $ c $ such that $ f(c)=0$.

The second way is to construct this kind of element by saying let $ c $ be ... or let $ c= $. For example when we want to prove the existence of a rational $ c $ between any two reals, we construct it from the Archimedean property.

hamam_Abdallah
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To prove $\exists x P(x)$, you can construct a specific element $c$ to satisfy $P$. This is a constructive proof. Another way is to use an indirect or non-constructive proof.

For an indirect proof, you often use proof by contradiction. Consider the tautology: $$( \neg p \rightarrow \bot) \rightarrow p$$ This rule means, if we assume the negation of proposition $p$ and derive a contradiction, it follows that $p$ is true.

Let $p$ be $\exists x P(x)$, an indirect proof will usually assume $\neg p$ which is $\forall x \neg P(x)$ and derive a contradiction. Once a contradiction is derived, it follows that $\exists x P(x)$.

For a non-constructive proof, an indirect proof can be used, but it does not have to be. Consider the beautiful problem: "can an irrational be raised to an irrational power to get a rational number?" The answer is yes and it uses proof by cases rather than straight contradiction. Take a look at this post for a concise discussion: Can an irrational number raised to an irrational power be rational? . This is a frustrating but interesting result that is non-constructive.

user512716
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