State the contraction mapping theorem.

A metric space $(X, d)$ is bounded if $\{d(x, y) \mid x, y \in X\}$ is a bounded subset of $\mathbb{R}$. Suppose $(X, d)$ is complete and bounded. Let $\operatorname{Maps}(X, X)$ denote the set of continuous $\operatorname{maps}$ from $X$ to itself. For $f, g \in \operatorname{Maps}(X, X)$, let

$$\delta(f, g)=\sup _{x \in X} d(f(x), g(x))$$

Prove that $(\operatorname{Maps}(X, X), \delta)$ is a complete metric space. Is the subspace $\mathcal{C} \subset \operatorname{Maps}(X, X)$ of contraction mappings a complete subspace?

Let $\tau: \mathcal{C} \rightarrow X$ be the map which associates to any contraction its fixed point. Prove that $\tau$ is continuous.