Question 63
Let $f : [0, \infty) \to [0, \infty)$ be a continuous function with $f(0) \gt 0$, such that for any $0 \le x \lt y$, we have
$$
x - y \lt f(y) - f(x) \le 0.
$$
Prove that:
a) there exists a unique number $\alpha \in (0, \infty)$ with the property $(f \circ f)(\alpha) = \alpha$;
b) the sequence $(x_n)_{n \ge 1}$, defined by $x_1 \ge 0$ and $x_{n+1} = f(x_n)$, for any $n \in \mathbb{N}^*$, is convergent.
$$
x - y \lt f(y) - f(x) \le 0.
$$
Prove that:
a) there exists a unique number $\alpha \in (0, \infty)$ with the property $(f \circ f)(\alpha) = \alpha$;
b) the sequence $(x_n)_{n \ge 1}$, defined by $x_1 \ge 0$ and $x_{n+1} = f(x_n)$, for any $n \in \mathbb{N}^*$, is convergent.