I hope this is advanced enough... 
We have a sequence of \(\displaystyle c_n=c^2_{n-1}-1\):
»Prove, that for any chosen number \(\displaystyle c_0\) on the interval \(\displaystyle (A,B)\), where \(\displaystyle A\) and \(\displaystyle B\) are x-coordinates of intersection points of functions \(\displaystyle f(x)=x^2-1\) and \(\displaystyle g(x)=x\) (both graphed below), all other terms of the sequence are on that same interval, and, that the sequence has two convergence points \(\displaystyle 0\) and \(\displaystyle -1\)«
(My ) reasoning: this is a recursively given sequence \(\displaystyle c_n=x^2-1\), whose first term, \(\displaystyle c_0\), when between \(\displaystyle A\) and \(\displaystyle B\), *should* always interchangeably give \(\displaystyle 0\) and \(\displaystyle -1\).
Chosing \(\displaystyle c_0=1\) obviously gives us convergence to \(\displaystyle 0\) and \(\displaystyle -1\): \(\displaystyle c_1=0\), \(\displaystyle c_2=-1\), \(\displaystyle c_3=0\), \(\displaystyle c_4=-1\), ...
I've calculated the two intersection points at \(\displaystyle A=\frac{1-\sqrt5}{2}\) and \(\displaystyle B=\frac{1+\sqrt5}{2}\).
What would prove convergence to \(\displaystyle 0\) and \(\displaystyle -1\) for any chosen \(\displaystyle c_0\in(A,B)\) :?:
We have a sequence of \(\displaystyle c_n=c^2_{n-1}-1\):
»Prove, that for any chosen number \(\displaystyle c_0\) on the interval \(\displaystyle (A,B)\), where \(\displaystyle A\) and \(\displaystyle B\) are x-coordinates of intersection points of functions \(\displaystyle f(x)=x^2-1\) and \(\displaystyle g(x)=x\) (both graphed below), all other terms of the sequence are on that same interval, and, that the sequence has two convergence points \(\displaystyle 0\) and \(\displaystyle -1\)«
(My
) reasoning: this is a recursively given sequence \(\displaystyle c_n=x^2-1\), whose first term, \(\displaystyle c_0\), when between \(\displaystyle A\) and \(\displaystyle B\), *should* always interchangeably give \(\displaystyle 0\) and \(\displaystyle -1\). Chosing \(\displaystyle c_0=1\) obviously gives us convergence to \(\displaystyle 0\) and \(\displaystyle -1\): \(\displaystyle c_1=0\), \(\displaystyle c_2=-1\), \(\displaystyle c_3=0\), \(\displaystyle c_4=-1\), ...
I've calculated the two intersection points at \(\displaystyle A=\frac{1-\sqrt5}{2}\) and \(\displaystyle B=\frac{1+\sqrt5}{2}\).
What would prove convergence to \(\displaystyle 0\) and \(\displaystyle -1\) for any chosen \(\displaystyle c_0\in(A,B)\) :?: