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.

$\blacksquare$