Define a contraction mapping and state the contraction mapping theorem.

Let $(X, d)$ be a non-empty complete metric space and let $\phi: X \rightarrow X$ be a map. Set $\phi^{1}=\phi$ and $\phi^{n+1}=\phi \circ \phi^{n}$. Assume that for some integer $r \geqslant 1, \phi^{r}$ is a contraction mapping. Show that $\phi$ has a unique fixed point $y$ and that any $x \in X$ has the property that $\phi^{n}(x) \rightarrow y$ as $n \rightarrow \infty$.

Let $C[0,1]$ be the set of continuous real-valued functions on $[0,1]$ with the uniform norm. Suppose $T: C[0,1] \rightarrow C[0,1]$ is defined by

$$T(f)(x)=\int_{0}^{x} f(t) d t$$

for all $x \in[0,1]$ and $f \in C[0,1]$. Show that $T$ is not a contraction mapping but that $T^{2}$ is.