User:Caliburn/s/1

Theorem
Let $\struct {X, d}$ and $\struct {Y, d'}$ be metric spaces.

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

Let $\sequence {f_n}_{n \in \N}$ be a uniformly convergent sequence in $\map {\mathcal C} {X, Y}$.

Let $\mathcal F = \set {f_i : i \in \N}$.

Then $\mathcal F$ is pointwise equicontinuous.