Uniformly Convergent Sequence Evaluated on Convergent Sequence

Theorem
Let $X = \left({A, d}\right)$ and $Y = \left({B, \rho}\right)$ be metric spaces.

Let $\mathcal F = \left \langle{f_n}\right \rangle$ be a uniformly convergent sequence of continuous mappings $f_n: X \to Y$.

Let $\left \langle{a_n}\right \rangle$ be a Convergent Sequence in $X$ with limit $a \in X$.

Then $\left \langle{f_n\left({a_n}\right)}\right \rangle$ is convergent and
 * $\displaystyle \lim_{n \to \infty} f_n \left({a_n}\right) = f(a)$.

Proof
We want to show that:
 * $\left|{ f_n\left({a_n}\right) - f(a) }\right| \to 0$ as $n \to \infty$.

From the Triangle Inequality:
 * $\left|{ f_n\left({a_n}\right) - f(a) }\right| \leq \left|{ f_n\left({a_n}\right) - f\left({a_n}\right) }\right| + \left|{ f\left({a_n}\right) - f(a) }\right|$.

Now fix $\eps \in \R^{> 0}$

Since $f$ is continuous and $a_n \to a$, Definition Sequential Continuity is Equivalent to Continuity in Metric Space tells us that:
 * $\exists M \in \N: n \geq \M \implies \left|{ f_n\left({a_n}\right) - f\left({a_n}\right) }\right| < \frac{\eps}{2}$

Since $\left \langle{f_n}\right \rangle$ is uniformly convergent:
 * $\exists N \in \N: n \geq \N \implies \left|{f_n\left({a_n}\right) \to f\left({a_n})}\right| < \frac{\eps}{2}$

So if $n \geq \max(M, N)$:
 * $\left|{ f_n\left({a_n}\right) - f(a) }\right| \leq \left|{ f_n\left({a_n}\right) - f\left({a_n}\right) }\right| + \left|{ f\left({a_n}\right) - f(a) }\right| < \frac{\eps}{2} + \frac{\eps}{2} = \eps$.

This completes the proof.