Existence of Transcendental Numbers
Theorem
There exist (uncountably many) transcendental numbers. That is, not every complex number is the root of a polynomial over the rational numbers.
Proof
Assume by way of contradiction that the set of transcendental numbers is countable. Then, because the set of algebraic numbers is countable, their union, that is, the set of complex numbers, must also be countable. However since the real and complex numbers are equinumerous, and the real numbers are uncountable, this is a contradiction.