Fixed point method : where is my error?

I've mostly solved the following problem :

The equation has a zero in .

a)Verify that the function has fixed points and that the derivative of g in a fixed point has its absolute value greater than . 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 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 closed to the fixed point of g in , analyze the convergence of the sequence . **This is where I'm stuck in. My work : **I chose . I get . Then . 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?