Question 56
Prove that for all $x \in [\frac{1}{111}, \frac{110}{111}]$ there exist numbers $a_i \in \{-1, 1\}, i = 1, 2, \dots, 101$ such that
$$
|x_{101} - x| \le \frac{1}{402},
$$
where
$x_0 = 1, \quad x_k = (x_{k-1} + 1)^{a_k}, \quad \text{for } k = 1, 2, \dots, 101.$
$$
|x_{101} - x| \le \frac{1}{402},
$$
where
$x_0 = 1, \quad x_k = (x_{k-1} + 1)^{a_k}, \quad \text{for } k = 1, 2, \dots, 101.$