Exercise 1.3 (10 pts) Suppose a sequence \{x_i\}_{i \in \mathbb{N}} in \mathbb{R}^n converges q-quadratically to $x$, i.e., there exists a constant $C \ge 0$ such that
$\|x - x_{k+1}\| = C \|x - x_k\|^2$
for all $k = 1, 2, \dots$. Show that there exist a natural number $N$ and a nonnegative number $C'$, depending only on $C$, such that for any $\epsilon > 0$,
$\|x - x_{N+T}\| \le \epsilon$
for all $T \ge \log \left( \log \left(\frac{C'}{\epsilon}\right) \right)$.
