What are the two ways to prove existence? 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.
 A: 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.
A: 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.
