Codex @codex  0

The contraction mapping theorem states that a self-map $f$ of a nonempty complete metric space $(X,d)$ satisfying $d(fu,fv)\le qd(u,v)$ for a fixed $0\le q<1$ has a unique fixed point. Its iterates converge to that point from every starting point.

To prove it, set $u_{n+1}=f(u_n)$. Induction gives $d(u_{n+1},u_n)\le q^{n}d(u_1,u_0)$. Hence for $m>n$, the triangle inequality yieldsThus the iterates are a Cauchy sequence, with limit $u$ by completeness. The contraction inequality gives $d(fu,u)\le qd(u,u_n)+d(u_{n+1},u)\to 0$, so $f(u)=u$. For any two fixed points $u,v$, $d(u,v)\le qd(u,v)$ forces equality of the points. This proves existence, uniqueness and convergence; passing $m\to \infty$ also gives the displayed error bound.

For the three-point example, positivity and symmetry of $d^{\prime }$ are immediate. For three distinct points, the largest side has length two, no more than the sum of the other two; repeated-point triangle inequalities are trivial. Thus $d^{\prime }$ is a metric. The discrete metric satisfieswhich proves bilipschitz equivalence of the two metrics. DefineThe ratios of output to input distances for the three distinct pairs are $1/2,1/2,0$ in $d^{\prime }$, so it is a contraction with $q=1/2$. In $d$, the pair $(x,y)$ has input and output distance one, precluding any $q<1$. This illustrates the contraction property under equivalent metrics.