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!

Assume that $f$ is a function defined in a certain interval $\Delta$ and you want to prove that $$\sum_{i=1}^n f(x_i)\geqslant nf(a),\,a=\frac1n \sum x_i,\tag 1$$ where the numbers $x_i$ in $\Delta$ are arbitrary or satisfy some additional condition U. Example of U would be '$x_i+x_j\leqslant 2b$ for $i\ne j$.' Then Cauchy induction allows to consider only the case $n=2$, which is often handy. In the above example we induct from $n$ to $n+1$ bit tricky: replace two largest numbers to the average and then add the total average.

Specific example: if $x_i$ are non-negative numbers and the sum of any two does not exceed $\pi$, then $\sum \cos x_i\leqslant n \cos a$.

In the case when there is no condition U, we have simply Jensen inequality, which gives AM-GM for $f=-\log$ or for $f=\exp$. Of course, for other convex functions $f$ it also works. Moreover, you may define, as Martin Sleziak writes in the comment, 'midpoint convexity' of $f$ by $f(x)+f(y)\geqslant 2f\left(\frac{x+y}2\right)$ (this is equivalent to convexity under additional assumptions like continuity), and deduce (1) by Cauchy induction.

This is due to Noga Alon, recently popularized by Gil Kalai. I started a big-list thread motivated by this proof, and the answer by Federico Poloni reminded about this question.

Theorem. Let $G=(V,E)$ be a bipartite $r$-regular multigraph. Then $E$ is a union of $r$ perfect matchings.

Proof. At first, we prove it when $r$ is a power of 2, by induction. Base $r=1$ is clear. If $r>1$ is even, the number of edges in each connected component is even (sum up degrees of one part.) Take an Eulerian cycle in every connected component and color edges alternatively, we partition $E$ onto two $r/2$-regular multigraphs. Apply induction proposition for them.

Now assume that $r$ is not a power of 2. Take large $N$ and write $2^N=rq+t$, $0<t<r$. Replace each edge of our multigraph onto $q$ edges, also add extra edges formed by arbitrary $t$ perfect matchings. This new multigraph may be partitioned onto $2^N$ perfect matchings, and if $N$ is large enough, some of them do not contain extra edges. So, we have found a perfect matching in initial graph, removing it and repeating $r$ times gives a required decomposition.