State and prove the contraction mapping theorem.

Let $A=\{x, y, z\}$, let $d$ be the discrete metric on $A$, and let $d^{\prime}$ be the metric given by: $d^{\prime}$ is symmetric and

$$\begin{gathered} d^{\prime}(x, y)=2, d^{\prime}(x, z)=2, d^{\prime}(y, z)=1 \\ d^{\prime}(x, x)=d^{\prime}(y, y)=d^{\prime}(z, z)=0 \end{gathered}$$

Verify that $d^{\prime}$ is a metric, and that it is Lipschitz equivalent to $d$.

Define an appropriate function $f: A \rightarrow A$ such that $f$ is a contraction in the $d^{\prime}$ metric, but not in the $d$ metric.