Method of Infinite Descent/Proof 1

Proof
Let $G$ be the set of all $n$ such that $\map P n$ hold.

That is:
 * $G = \set {n \in \N : \map P n}$

suppose that $G$ is non-empty.

By the Well-Ordering Principle $G$ contains a smallest element $n_\alpha$.

But by the descent step, there exists $n_\beta \in \N$ that strictly precedes $n_\alpha$ for which $\map P {n_\beta}$.

But then $n_\beta \in G$, which contradicts $n_\alpha$ being the smallest element.

Therefore $G$ is empty.

That is $\map P n$ is false for every $n$.