Let $y_1, y_2 >0$ and $y_{n}=\frac{2}{y_{n-1}+y_{n-2}}$ for $n\ge 3$. I know that this sequence is converging and that $y_n \to 1$ as $n\to \infty$. Let us write $Y_n = (y_n,y_{n+1})$ and $f(a,b)=(b,\frac{2}{a+b})$ so that $f(Y_n)=Y_{n+1}$. My question is the following:
Is it possible to find a metric on $\mathbb{R}_+ \times \mathbb{R}_+$ such that $f$ is a contraction mapping?
An idea would be to look at $\nabla f = \begin{bmatrix}0 & -\frac{2}{(a+b)^2} \\ 1 & -\frac{2}{(a+b)^2}\end{bmatrix}$ and make it small in some appropriate norm (that is to be constructed), but I have no clue how to do it. Then if we do that, an application of the mean-value theorem would give the result.