Convergence of given sequence Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function with:

$$f(x) = x - \arctan{x}$$

We consider the sequence $(x_{n})$ with $x_{0} &gt; 0$ and $x_{n + 1} = f(x_{n})$, for any $n \in \mathbb{N}$.

Prove that $(x_{n})$ is convergent and find its limit.

So far, I've proved that $f(x) \geq 0$ for any $x \geq 0$. From this I've concluded that $(x_{n})$ is a positive sequence. Now I need to find $(x_{n})$'s monotony. I've calculated both $x_{n + 1} - x_{n}$ and $\frac{x_{n + 1}}{x_{n}}$. For the first I've got $- \arctan{x_{n}}$ and the second one gave me $1 - \frac{\arctan{x_{n}}}{x_{n}}$. From this point I don't know what to do next.

Thank you in advance!
Look at the graph: (from Wolfram Alpha)



The way to answer this question is to prove that, say for $x &gt; 0$, $f(x) &lt; x \le 0$. It should be clear that iterating the function only gets you closer to zero.

Does this help?
The derivative is $\frac{x^2}{1+x^2}$and  the function is a contraction mapping on any interval $[0,n]$. The given sequence is used to find the  fixed point x=0.