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 $\sequence {f_n}$ be a sequence of mappings $f_n : X \to Y$.

Suppose that:
 * $(1): \quad \sequence {f_n}$ is uniformly convergent on $S$
 * $(2): \quad \ds \forall n \in \N : \lim_{x \mathop \to c} \map {f_n} x$ exists

Then:
 * $\ds \lim_{x \mathop \to c} \lim_{n \mathop \to \infty} \map {f_n} x = \lim_{n \mathop \to \infty} \lim_{x \mathop \to c} \map {f_n} x$