A function is called a contraction if there is a number $c$ with $0 < c < 1$ so that $|f(x) − f(y)| ≤ c|x − y|$ for all $x, y ∈ \mathbb{R}$.

Question

Real Analysis:

A function $f : \mathbb{R} \rightarrow \mathbb{R}$ is called a contraction if there is a number $c$ with $0 < c < 1$ so that $|f(x) − f(y)| ≤ c|x − y|$ for all $x, y ∈ \mathbb{R}$. Let $f(x)$ be a contraction.

I already proved that $f(x)$ is continuous and that the sequence $y_n$ is Cauchy so hence it converges to some limit $y$. I also already showed that the value $y$ is a fixed point of $f$. i.e., $f(y)=y$.

Now I need to do this last part:

(d) Show that for any $x \in \mathbb{R}$ we have $lim_{n\rightarrow \infty} f^n(x)=y$ , where $f^n$ is $f$ applied $n$ times.

Answer 1

Hint: using $f^n(y)=y$ for all $n$, we have $$|f^n(x) - y| \le c |f^{n-1}(x) - y| \le c^2 |f^{n-2}(x) - y| \le \cdots \le c^n |x-y|.$$