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:

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

The validity of the material on this page is questionable. In particular: The left hand side may diverge, since $\lim_{n \mathop \to \infty} \map {f_n} x $ is not assumed to converge for $x \not \in S$. Overall $S$ does not seem effectively used in the statement. Some condition is missing. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Questionable}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Source of Name

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