Exercise 1.3 (10 pts) Suppose a sequence {x_k} in R converges q-quadratically to x, i.e., there exists a constant C > 0 such that
||x_{k+1} - x|| = C|x_k - x|^2
for all k = 1,2,.... Show that there exist a natural number N and a nonnegative number C' depending only on C, such that for any ε > 0,
|x - x_{N+T}| ≤ C'ε
for all T ≥ log(log(1/ε))