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?
Best Answer
Many existence proofs which exploit the idea of Baire category.
For instance, existence of a metrically transitive automorphism of the closed unit square was first obtained by the category method (see "Measure-preserving homeomorphisms and metrical transitivity" by Oxtoby and Ulam) . Another classical example is due to Banach who proved that every function from a residual subset of $C[0,1]$ is nowhere differentiable.
A nice and elementary book by Oxtoby discusses these and many other applications of the category method.