Method of Infinite Descent

Theorem
Let $\map P {n_\alpha}$ be a propositional function depending on $n_\alpha \in \N$.

Let $\map P {n_\alpha} \implies \map P {n_\beta}$ such that $0 < n_\beta < n_\alpha$.

Then we may deduce that $\map P n$ is false for all $n \in \N$.

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

This technique is known as the method of infinite descent.

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