SCHWEITZER 2021 Problem 5

Let $f(x)=\frac{1+\cos(2 \pi x)}{2}$ , for $x \in \mathbb{R}$ , and $f^n=\underbrace{ f \circ \cdots \circ f}_{n}$ . Is it true that for Lebesgue almost every $x$ , $\lim_{n \to \infty} f^n(x)=1$ ?