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?