Constructive proof

A constructive proof demonstrates the existence of a mathematical function, number or object by producing (constructing) it. This is in contrast with other styles of proof, such as proof by contradiction, which asserts the existence of an object by finding a contradiction if it did not exist. Such a proof is called nonconstructive.

The Axiom of Choice assumes the existence of a function without constructing it, and thus all proofs that rely on the Axiom of Choice are nonconstructive proofs.

The easiest way to prove the existence of transcendental numbers is by a nonconstructive proof, arguing that the set of real numbers is uncountable while the set of algebraic numbers is countable, and thus (many) transcendental numbers must exist. Of course, finding a specific example is a much more difficult endeavor.