User:Caliburn/s/fa/Arzelà-Ascoli Theorem/Corollary

Theorem
Let $\struct {X, d}$ be a compact metric space.

Let $\map C {X, \R}$ be the set of continuous functions from $X$ to $\R$.

Let $\norm \cdot_\infty$ be the supremum norm on $\map C {X, \R}$.

Let $d'$ be the metric induced by $\norm \cdot_\infty$.

Let $\struct {A, d'_A}$ be a metric subspace of $\struct {\map C {X, \R}, d'}$.

Then Further, $\struct {A, d'_A}$ is a compact metric space :


 * $(1): \quad$ $A$ is closed
 * $(2): \quad$ there exists a real number $R > 0$ such that $\norm f_\infty \le R$ for all $f \in A$
 * $(3): \quad$ $A$ is uniformly equicontinuous in $\struct {\map C {X, \R}, d'}$.

Proof
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