The proof uses two facts. Fact A: a polynomial satisfies on for some constant , obtained by factoring each difference of powers . Fact B: a non-zero polynomial of degree has at most real roots.

Write , so that and for some natural . If for a polynomial of degree with integer coefficients, then for all large we have (Fact B), and for some non-zero integer , so

Since grows faster than , this fails for large : contradiction.

Numbers of this form are called Liouville numbers. The proof shows more generally that any real with

is transcendental: in loose terms, a real number with very good rational approximations is transcendental. The same proof does not apply to , but it is known that is transcendental as well.