Analysis II | Part IB, 2001

State and prove the contraction mapping theorem.

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

d(x,y)=2,d(x,z)=2,d(y,z)=1d(x,x)=d(y,y)=d(z,z)=0\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 dd^{\prime} is a metric, and that it is Lipschitz equivalent to dd.

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

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