Fixed point method : where is my error?

I've mostly solved the following problem :

The equation $\displaystyle x^2-x-1=0$ has a zero in $\displaystyle [-1,0]$.

a)Verify that the function $\displaystyle g(x)=x^2-1$ has $\displaystyle 2$ fixed points and that the derivative of g in a fixed point has its absolute value greater than $\displaystyle 1$. Take note that the existence of a fixed point doesn't imply that the derivative evaluated in this point has an absolute value lesser than 1. **DONE**

b)Show that $\displaystyle g(x)$ satisfies the conditions of existence of a fixed point in [-1,0]. **DONE **(I showed that g was contractive on this interval)

c)Chose a $\displaystyle x_0$ closed to the fixed point of g in $\displaystyle [-1,0]$, analyze the convergence of the sequence $\displaystyle x_{n+1}=g(x_n)$. **This is where I'm stuck in. My work : **I chose $\displaystyle x_0=-1$. I get $\displaystyle x_1=0$. Then $\displaystyle x_2=-1$. It shouldn't be possible! It's not convergent, it will oscillate infinitely! Where is my error? If g is contractive on [-1,0], it must have a fixed point, so what am I doing wrong here?