The recursion theorem
In set theory, this is a theorem guaranteeing that recursively defined functions exist. Given a set $X$, $a \in X$ and a function $f \colon X \to X$, the theorem states that there is a unique function $F:\mathbb{N} \to X$ (where $\mathbb{N}$ denotes the set of natural numbers including zero) such that
$$F(0) = a$$
$$F(n + 1) = f(F(n))$$
for any natural number $n$.
How do you apply this in practice? How, for example, do you apply it to prove the existence of the factorial function?