The proof relies on infinite descent, and is only briefly sketched in the letter.

This is the step that uses infinite descent.

Since an infinite descent is impossible, we conclude that must be expressible as a sum of two squares, as claimed.

One proof of the number's irrationality is the following proof by infinite descent.

It is also an example of proof by infinite descent.

The classic proof that the square root of 2 is irrational operates by infinite descent.

Therefore, by the argument of infinite descent, the original solution (x, y, z) was impossible.

He did make repeated use of mathematical induction, introducing the method of infinite descent.

For other proofs of this by infinite descent, see and.

The first proof was found by Euler after much effort and is based on infinite descent.