# Abstract Nonsense

## Simple Fixed Point Problem

This is just a simple fixed point problem I did on mathhelpforum a while ago. It’s not particularly difficult.

Problem: Let $\mathcal{M}$ a compact metric space and $\varphi:\mathcal{M}\to\mathcal{M}$ a contractive mapping. Then, $\varphi$ has a unique fixed point.

Proof: Clearly the mapping $\text{dist}:\mathcal{M}\to\mathbb{R}:x\mapsto d(x,\varphi(x))$ is continuous since $\text{dist}=d\circ\left(\text{id}\oplus\varphi\right)$. Thus, by the compactness of $\mathcal{M}$ we have that $\text{dist}$ attains a minimum at some point $x_0\in\mathcal{M}$. Now, suppose that $\varphi(x_0)\ne x_0$, then $d\left(\varphi(\varphi(x_0)),\varphi(x_0)\right) which contradicts the minimality of $x_0$. It follows that $x_0$ is in fact a fixed point. Now, clearly it is unique for if we supposed that $\varphi(x_1)=x_1,\text{ }x_1\ne x_0$ then we’d have that

$d(x_0,x_1)=d(f(x_0),f(x_1))

The conclusion follows. $\blacksquare$