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?



On  proving that a certain set is not empty by proving that it is actually large
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"](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=oxtoby&s5=metrical%2520transitivity&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq) 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](http://books.google.co.uk/books?id=wUDjoT5xIFAC&pg=PR7&lpg=PR7&dq=measure+and+category&source=bl&ots=4EMy1X1RUN&sig=cd5KeoXCfve3gNwvk2wyrOmTLmU&hl=en&ei=VS5YTIfsH9GT4gaesIj_Bg&sa=X&oi=book_result&ct=result&resnum=5&ved=0CDIQ6AEwBA#v=onepage&q&f=false) by Oxtoby discusses these and many other applications of the category method.


