Method of Infinite Descent

Theorem
Let $$P \left({n_\alpha}\right)$$ be a propositional function depending on $$n_\alpha \in \N$$.

Let $$P \left({n_\alpha}\right) \implies P \left({n_\beta}\right)$$ such that $$0 < n_\beta < n_\alpha$$.

Then we may deduce that $$P \left({n}\right)$$ is false for all $$n \in \N$$.

That is, suppose that by assuming the truth of $$P \left({n_\alpha}\right)$$ for any natural number $$n_\alpha$$, we may deduce that there always exists some number $$n_\beta$$ strictly less than $$n_\alpha$$ for which $$P \left({n_\beta}\right)$$ is also true, then $$P \left({n_\alpha}\right)$$ can not be true after all.

This technique is known as the method of infinite descent.

The process of deducing the truth of $$P \left({n_\beta}\right)$$ from $$P \left({n_\alpha}\right)$$ such that $$0 < n_\beta < n_\alpha$$ is known as the descent step.

Proof
Suppose that $$P \left({n_\alpha}\right)$$ holds.

Then from the descent step, $$\exists n_\beta \in \N_{n_\alpha}: P \left({n_\beta}\right)$$.

The descent step then tell us we can deduce a smaller positive solution, $$n_\gamma$$, such that $$P \left({n_\gamma}\right)$$ is true and $$n_\gamma \in \N_{n_\beta}$$.

And again, the descent step tells us we can deduce a still smaller positive solution, $$n_\delta$$, such that $$P \left({n_\delta}\right)$$ is true and $$n_\delta \in \N_{n_\gamma}$$.

Now, consider the unending sequence: $$n_\alpha > n_\beta > n_\gamma > n_\delta > \cdots > 0$$.

The set $$S = \left\{{n_\alpha, n_\beta, n_\gamma, n_\delta, \ldots}\right\}$$ is not bounded below, as for any $$\forall x \in S: \exists y \in S: y < x$$.

By the well-ordering principle, any non-empty subset of $$\N$$ must have a least element.

As $$S$$ is not bounded below, it has no least element, so must be empty.