Moore-Osgood Theorem

Theorem
Let $X$ and $Y$ be metric spaces.

Let $S$ be a subspace of $X$.

Let $c$ be a limit point of $S$.

Let $\left \langle{f_n}\right \rangle$ be a sequence of mappings $f_n : X \to Y$.

Suppose that:
 * $(1): \quad \left \langle{f_n}\right \rangle$ is uniformly convergent on $S$
 * $(2): \quad \displaystyle \forall n \in \N : \lim_{x \to c} f_n \left({ x }\right)$ exists

Then:
 * $\displaystyle \lim_{x \to c} \lim_{n \to \infty} f_n \left({ x }\right) = \lim_{n \to \infty} \lim_{x \to c} f_n \left({ x }\right)$