I want just how to solve it give me a hint

The question:-

Use fixed-point iteration method to determine a solution accurate to within $\displaystyle 10^{-2}$ for $\displaystyle x^3-x-1=0 $ on [1,2] use $\displaystyle p_{0}=1 $

here is my solution (correct me if I was wrong )

first we should dtermine F(x)=x so

$\displaystyle x^3-x-1=0 \Rightarrow x^3=x+1 \Rightarrow x=\sqrt[3]{x+1}\Rightarrow F(x)=\sqrt[3]{x+1}$

$\displaystyle F'(x)=\frac{1}{3\sqrt[3]{(x+1)^2}}$

$\displaystyle \mid F'(x) \mid <1 $ so F(x) is converge now

$\displaystyle F(p_{0})=\sqrt[3]{2}=1.2599210498949=p_1$

$\displaystyle F(p_1)=1.3122938366833=p_2 $

$\displaystyle F(p_2)=1.3223538191388=p_3 $

..................etc

when I should stop ??

my solution is correct or it is wrong ??

Thanks for the help