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 $\sequence {f_n}$ 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.