Fixed-point iteration, Convergence of a sequence?

Given is the function $f(x)=x^{3}+x-1$ on $\mathbb{R}$. Use the Fixed-point iteration for $x\in \left [ 0.5 , 1 \right ]$ to show that the sequence $\left \{ x \right \}_{n}$ converges to the fixed-point $s$ for $n\rightarrow + \infty$.

Of course, i can write $f(x)$ as $\frac{1}{1+x^{2}}=x$. I also show that the maximum of the function in this interval is $0,64$, after evaluation of the derivative of first order. I apply directly the fixed-point iteration with $x_{0}=0.5$ and get $x_{1}=0.8$, $x_{2}=0.61$, $x_{3}=0.729$, $x_{4}=0.653$ etc, with using software i calculated $s=0.682388$.

Here starts my problem. How can i prove formal that the sequence converges to $s$?

On the other side, i could appy the Banach fixed-point theorem, i know that $\mathbb{R}$ is a complete metric space with the absolute value norm and $\frac{1}{1+x^{2}}$ is a contraction. So, the sequence converges to a unique fixed point.

Is this correct? Can anybody help me, please?