How did infinite descent prove that no integer right triangle can have a square area?

To prove that no right triangle with integer sides can have an area equal to a perfect square, Pierre de Fermat employed a logical technique known as infinite descent.1

This method functions as an inverted form of mathematical induction. The proof follows these steps:2

• Assumption: The argument begins by assuming that there exists at least one right triangle with integer sides ($x,y,z$) whose area ($w^2=\frac{xy}{2}$) is a perfect square.1
• Generation: Using algebraic manipulation, Fermat demonstrated that if such a triangle exists, one could construct a new, smaller right triangle that also possesses an integer-square area.1
• Contradiction: By reapplying this process, one generates an endless sequence of strictly decreasing positive integers.1
• Conclusion: Because any set of positive integers must contain a smallest member, it is impossible to have an infinite sequence of decreasing positive integers. This contradiction proves that the initial assumption must be false; therefore, no such integer right triangle can exist.1