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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.

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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 $k$. The is a "book" proof according to Erdos.

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hi Hans and thanks for your reply. – Pi314 May 17 '13 at 13:04
hi Hans and thanks for your reply. I understand what ascending and descending induction separately are. However, its use in Lang (or better page 79 of Silverman's "The Arithmetic of Elliptic Curves", 2nd ed) suggests(?) something else. Or maybe I am not seeing correctly what they are inducting on in both directions. – Pi314 May 17 '13 at 13:11
what is the context? – Hans Engler May 17 '13 at 21:43

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