Moore-Osgood Theorem

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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:

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


Proof


Source of Name

This entry was named for Eliakim Hastings Moore and William Fogg Osgood.