if $\left | x_{n+1}-\frac{x_{n}^2}{x_{n-1}} \right |\leq 1$, show that $(\frac{x_{n+1}}{x_{n}}) $ convergent
What about the following sequence: $$1,2,3,\frac{7}{2},\frac{37}{12},\frac{865}{504},1,1,\frac{1}{2},1,1,\frac{1}{2},1,1,\frac{1}{2},...$$
I believe it satisfies the given conditions, but clearly $\frac{x_{n+1}}{x_n}$ doesn't converge. I think the problem statement might be wrong.