What is a contraction map on a metric space $X$ ? State and prove the contraction mapping theorem.

Let $(X, d)$ be a complete non-empty metric space. Show that if $f: X \rightarrow X$ is a map for which some iterate $f^{k}(k \geqslant 1)$ is a contraction map, then $f$ has a unique fixed point. Show that $f$ itself need not be a contraction map.

Let $f:[0, \infty) \rightarrow[0, \infty)$ be the function

$$f(x)=\frac{1}{3}\left(x+\sin x+\frac{1}{x+1}\right)$$

Show that $f$ has a unique fixed point.