I want to show that the sequence given by $$x_{n+1} = Tx_n = x_n-\frac{(x_n^2-2)}{x_n+x_{n-1}}$$ forms a contraction mapping. That is $$|Tx_1-Tx_2|\leq c|x_1-x_2|.$$ Where $c$ is to be determined. I am also unsure of what the conditions for convergence are. So far I have
$$\begin{align*}|Tx_1-Tx_2| &= \bigg|(x_1-\frac{(x_1^2-2)}{x_1+x_0})-(x_2-\frac{(x_2^2-2)}{x_2+x_1})\bigg| \\ &= \bigg|\frac{(x_1^2-2)(x_0-x_2)}{(x_1+x_0)(x_1+x_2)}\bigg| \end{align*}$$ and I am not really sure where to go from here.
I have also thought about $$|Tx_1-Tx_2| = \bigg|(x_1-x_2)-\frac{(x_1^2-2)}{x_1+x_0}+\frac{(x_2^2-2)}{x_2+x_1}\bigg|$$ and factoring out the $(x_1-x_2)$ term but I am also unsure of where to go from there.
I should note that this sequence comes from the secant method with $f(x)=x^2-2$. Would it be better to show that the secant method forms a contraction mapping? If so how would one do that?
Any help and comments would be appreciated. Thank you.