MathNet 2006 — Part 03 Question 153

Suppose $f(x) = x^2 + a$. Define $f^1(x) = f(x)$, $f^n(x) = f(f^{n-1}(x))$, $n = 2, 3, \dots$, and let $M = \{a \in \mathbb{R} \mid |f^n(0)| \le 2$, for any $n \in \mathbb{N}\}$. Prove that $M = [-2, \frac{1}{4}]$.