On page 88 of Lang's "Topics in Cohomology of Groups", Lang mentions a technique he calls "ascending and descending induction".
Initially I felt a bit embarrassed that I did not know a sort of induction, but then when googling for a definition, I found very few mentions of it and none that helped me understand exactly what this sort of induction is.
Could someone explain to me what ascending and descending induction is, please?
A reference to its definition in the literature would also be very helpful.
Let $(P_n)_{n \ge 1}$ be a set of statements. You want to prove that all $P_n$ are true.
In ascending induction, you prove (i) $P_1$ is true and (ii) $\forall n P_n \Rightarrow P_{n+1}$.
In descending induction, you prove (i) $P_n$ is true for infinitely many $n$ and (ii) $\forall n P_{n+1} \Rightarrow P_n$.
The best known example for descending induction is Cauchy's proof of the arithmetic-geometric mean inequality. Part (i) is done by proving that $P_n$ is true for all $n = 2^k$, which in turn is done by ascending induction over $k$. This is a "book" proof according to Erdos.