If sequence $A_n \rightarrow 0$ as $n \rightarrow \infty$, and the Shanks Transformation of $A_n$ defined as $$S\left(A_n\right)=\frac{A_{n+1}A_{n-1}-A_n^2}{A_{n+1}+A_{n-1}-2A_n}$$ also converges, prove $S\left(A_n\right)$ converges to zero.
It's problem 8-1 in Carl M. Bender's book, Advanced Mathematical Methods for Scientists and Engineers. I tried the definition of limit but couldn't finish the proof.
Equivalently $S(A_n)=A_{n-1}-\frac{(\Delta A_{n-1})^2}{\Delta^2 A_{n-1}}$ with the forward difference operator $\Delta u_n:=u_{n+1}-u_n$. (Note the problem statement should have included $\Delta^2 A_n\ne 0$.) Thus $$\frac{S(A_n)}{A_{n-1}}=1-\frac{\Delta A_{n-1}}{A_{n-1}}\cdot\frac{\Delta A_{n-1}}{\Delta^2A_{n-1}}.$$By Stolz–Cesàro this has limit $1-1=0$, so $S_n$ has limit $0\times 0=0$.