section_path: Methods of proof/Nonconstructive proof
section_name: Nonconstructive proof
paragraph_text: Nonconstructive proof
  
  A nonconstructive proof establishes that a mathematical object with a certain property exists—without explaining how such an object can be found. Often, this takes the form of a proof by contradiction in which the nonexistence of the object is proved to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. The following famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that a b {\displaystyle a^{b}} is a rational number. This proof uses that 2 {\displaystyle {\sqrt {2}}} is irrational (an easy proof is known since Euclid), but not that 2 2 {\displaystyle {\sqrt {2}}^{\sqrt {2}}} is irrational (this is true, but the proof is not elementary).