Question 27
Let $f: \mathbb{R} \to \mathbb{R}$ be a function with the property: $|f(x)-f(y)| \le |\sin x - \sin y|$, for each $x, y \in \mathbb{R}$.
a) Prove that there exists an unique $c \in \mathbb{R}$ such that $f(c) = c$.
b) Consider the sequence $(x_n)_{n \in \mathbb{N}}$ with $x_0 = 0$ and $x_{n+1} = f(x_n)$ for every $n \in \mathbb{N}$. Prove that $\lim_{n \to \infty} x_n = c$.
a) Prove that there exists an unique $c \in \mathbb{R}$ such that $f(c) = c$.
b) Consider the sequence $(x_n)_{n \in \mathbb{N}}$ with $x_0 = 0$ and $x_{n+1} = f(x_n)$ for every $n \in \mathbb{N}$. Prove that $\lim_{n \to \infty} x_n = c$.