TITLE: On  proving that a certain set is not empty by proving that it is actually large
QUESTION [12 upvotes]: It happens occasionally that one can prove that a given set is not empty by proving that it is actually large. The word "large" here may refer to different properties.
For example, one can prove that a certain set is not empty by proving that its cardinality is big, as in the proof that there exist transcendental numbers : The set of algebraic numbers is countable, but the set of real numbers is uncountable, so there is uncountably many transcendental numbers.
One could also prove that a certain set is not empty by proving, for example, that it has positive measure, that it is dense, etc.
What are some good examples of such proofs?

REPLY [3 votes]: The theme is prevalent in combinatorial number theory and ergodic theory. Consider the ergodic Szemerédi theorem, for instance. It says that if $(X,\mathcal{B},\mu,T)$ is a measure-preserving system and $A \in \mathcal{B}$ has $\mu (A) >0$, then $\forall k \in \mathbb{N}$ $\exists n \in \mathbb{N}$ so that $\mu (A \cap T^{-n}A \cap T^{-2n}A \cap \cdots \cap T^{-kn}A)>0$. It turns out that it's easier to prove that measure of the intersection is positive for infinitely many $n$. In fact, Furstenberg's original proof showed that $\liminf_{N \to \infty} \frac{1}{N} \sum_{n=1}^{N} \mu (A \cap T^{-n}A \cap T^{-2n}A \cap \cdots \cap T^{-kn}A) > 0$.