The proof by Cauchy induction of the arithmetic/geometric-mean inequality is well known. I am looking for a further theorem whose proof is much neater by this method than otherwise.



Is Cauchy induction used for proofs other than for AM–GM?
A nice proof by Cauchy induction can be given for the identity
$$ \|A^n\|=\|A\|^n, $$
which holds for a bounded, self-adjoint operator $A:H\to H$ on a real Hilbert space $(H,\langle\cdot,\cdot\rangle)$. Here $\|\cdot\|$ denotes the operator norm.


Indeed, the inequality $\|A^n\|\le\|A\|^n$ is trivial by submultiplicativity of such norm. Let us turn to the converse inequality: if you know that $\|A^n\|\ge\|A\|^n$, then
$$ \|A^{n-1}\|\|A\|\ge\|A^n\|\ge\|A\|^n, $$
so the same holds with $n-1$ in place of $n$. Thus it suffices to show that, whenever it holds for $n$, it holds also for $2n$:
$$ \|A^{2n}\|=\sup\_{\|x\|\le 1}\|A^{2n}x\|\ge\sup\_{\|x\|\le 1}\langle A^{2n}x,x\rangle=\sup\_{\|x\|\le 1}\langle A^nx,A^nx\rangle=\|A^n\|^2\ge\|A\|^{2n}. $$


This last line used Cauchy induction (i.e. the idea to prove $n\Rightarrow 2n$ rather than, e.g., $n\Rightarrow n+1$) in an essential way!


