Finite Set of Continuous Functions between Metric Spaces is Pointwise Equicontinuous

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 $\mathcal F = \set {f_1, f_2, \ldots, f_n}$ be a finite subset of $\map {\mathcal C} {X, Y}$.

Then $\mathcal F$ is pointwise equicontinuous.

Proof
Let $x \in X$.

Let $\epsilon > 0$.

Let $i$ be a natural number with $1 \le i \le n$.

Since $f_i$ is continuous at $x$, there exists $\delta_i > 0$ such that whenever:


 * $\map d {x, y} < \delta_i$

we have:


 * $\map {d'} {\map {f_i} x, \map {f_i} y} < \epsilon$

Let:


 * $\ds \delta = \min_i \set {\delta_i}$

so that:


 * $\delta \le \delta_i$

for each $i$.

Then, whenever:


 * $\map d {x, y} < \delta$

we have:


 * $\map {d'} {\map {f_i} x, \map {f_i} y} < \epsilon$

for each $i$.

So $\mathcal F$ is pointwise equicontinuous.