The existence of transcendental numbers can be shown easily by considering the cardinality of the set of solutions to polynomials with integer cofficents and the cardinality of the real numbers.
It is written here that http://en.wikipedia.org/wiki/Transcendental_number Liouville first proved the existence of transcendental numbers in 1844. I doubt that he proved this result by a cardinality argument, as I think that these ideas were first brought by Cantor and this was well after 1844. How did Liouville prove the existence of transcendental numbers.
He came up with the idea of what we now call a Liouville number, and then showed that all of them are transcendental.