Assume $K$ is a compact metric space with metric $\rho$ and $A$ is a map from $K$ to $K$ such that $\rho (Ax,Ay) < \rho(x,y)$ for $x\neq y$. Prove A have a unique fixed point in $K$. The uniqueness is easy. My problem is to show that there a exist fixed point. $K$ is compact, so every sequence h...