The proof is by contradiction. Suppose we returned to 440 after, say, n steps, and that in r of these n steps we had to divide by 2 the frequency we got, in order to stay within the range 440–880 (as we did when we replaced 990 with 495).

This situation is described by the equation

which may be written as

But this is a contradiction to the Fundamental Theorem of Arithmetic – also called the unique factorization theorem – which (roughly) says that every positive integer can be written uniquely as a product of prime numbers.