# 1617-257/TUT-R-8

Problem 1. Let ${\displaystyle E}$ be an infinite subset of a compact metric space ${\displaystyle X}$. Show that ${\displaystyle E}$ has a limit point.
Proof. If ${\displaystyle E}$ has no limit points, then ${\displaystyle E}$ is a closed subset of a compact space and is therefore compact in itself. Since each point of ${\displaystyle E}$ is isolated, we may find for each point ${\displaystyle e\in E}$ a neighborhood ${\displaystyle U_{e}}$ such that ${\displaystyle E\cap U_{e}=\{e\}}$. The collection ${\displaystyle \{U_{e}\}_{e\in E}}$ is an open cover of E which clearly has no finite subcover.
Problem 2. Let ${\displaystyle f:B_{1}(0)\to \mathbb {R} ^{2}}$ be a function which is "jelly-rigid": for all ${\displaystyle x,y\in B_{1}(0)}$: ${\displaystyle |f(x)-f(y)-(x-y)|\leq 0.1|x-y|}$. Prove that ${\displaystyle f}$ maps onto ${\displaystyle B_{0.4}(0)}$.
Proof. Since ${\displaystyle f}$ is Lipschitz, it has a unique extension to its boundary and so we regard it as a function on ${\displaystyle B:={\overline {B_{1}(0)}}}$. A simple estimate shows if ${\displaystyle f(x)\in B_{0.4}(0)}$, then ${\displaystyle x\in B_{1}(0)}$. Suppose now that there is some point ${\displaystyle z\in B_{0.4}(0)}$ which is not in the image of ${\displaystyle f}$. Let ${\displaystyle x_{0}\in B_{1}(0)}$ be a closest element to ${\displaystyle z}$ in the image of ${\displaystyle f}$. Consider now the point ${\displaystyle x_{1}:=x_{0}+\delta (z-f(x_{0}))}$ (jelly-rigidity says that the function is almost like the identity, so moving closer to the point ${\displaystyle z}$ from ${\displaystyle f(x_{0})}$ in the codomain side can be obtained by moving in that same direction on the domain side first) where ${\displaystyle \delta >0}$ is chosen to be small enough so that ${\displaystyle x_{1}\in B_{1}(0)}$ and ${\displaystyle 0<\delta <1}$. Then ${\displaystyle f(x_{1})}$ is closer to ${\displaystyle z}$ than is ${\displaystyle f(x_{0})}$.