Consider the sequence $\left \langle x_n \right \rangle,\; n \geq 0$ defined by the recurrence relation $x_{n + 1} = c \cdot (x_n)^2 - 2$, where $c > 0$.
For which of the following values of $c$, does there exist a non-empty open interval $(a, b)$ such that the sequence $x_n$ converges for all $x_0$ satisfying $a < x_0 < b$?
- $0.25$
- $0.35$
- $0.45$
- $0.5$
- i only
- i and ii only
- i, ii and iii only
- i, ii, iii and iv