Proof by infinite descent

Summary

A proof by infinite descent shows that a statement has no solution in positive integers. It assumes a solution, and from it constructs a strictly smaller solution in positive integers. This gives an infinite decreasing sequence of positive integers, which is impossible.

Remarks

The method rests on the well-ordering of the natural numbers: each nonempty set of natural numbers has a least element. A classic use shows that $\sqrt{2}$ is irrational: a solution of $a^2 = 2 b^2$ in positive integers gives a smaller one, $(2b - a, a - b)$. Fermat used the method to prove Fermat's right triangle theorem.

Explanations

L1 human

You cannot go down forever in the positive integers: you reach the bottom after finitely many steps. So if each solution gives a smaller one, there can be no solution at all.