I have a sequence: $x_{n+2} = \sqrt{x_{n+1}x_n}$, with $0<a\leq x_1 \leq x_2 \leq b$
I know that $ |x_{n+1}-x_n|=\frac{\sqrt{b}}{\sqrt{a}+\sqrt{b}}|x_n-x_{n-1}| $ or $|x_{n+1}-x_n|=(\frac{\sqrt{b}}{\sqrt{a}+\sqrt{b}})^{n-1}|x_2-x_1|$, and that the limit of this expression $\rightarrow0$ as $n\rightarrow\infty$, but I can't really figure out how to approach the $n>m$ part of this proof. If someone could give some pointers it would be greatly appreciated.
On
Once you have $|x_{n+1}-x_n|\leq c q^n$ with $c>0$ and $0<q<1$, you can telescope. This means $$ |x_{n+k}-x_n|=|\sum_{j=1}^kx_{n+j}-x_{n+j-1}|\leq \sum_{j=1}^k|x_{n+j}-x_{n+j-1}|=\sum_{j=1}^kcq^{n+j-1}=\frac{cq^n(1-q^k)}{1-q}\leq\frac{cq^n}{1-q}. $$ Now given $\varepsilon>0$, if you choose $n$ such that $\frac{cq^n}{1-q}<\varepsilon$, you get $|x_m-x_n|<\varepsilon$ for all $m\geq n$.
Hint : Firstly, go for m>n and use $|x_m - x_n| \le |x_m - x_{m-1}|+|x_{m-1} - x_{m-2}|+....+|x_{n+1} - x_n|$. Now plug in all the values.