Dini's Theorem
Jump to navigation
Jump to search
This theorem requires a proof..
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
Theorem
Let $K \subseteq \R$ be compact.
Let $\sequence {f_n}$ be a sequence of continuous real functions defined on $K$.
Let $\sequence {f_n}$ converge pointwise to a continuous function $f$.
Suppose that:
- $\forall x \in K : \sequence {\map {f_n} x}$ is monotone.
Then the convergence of $\sequence {f_n}$ to $f$ is uniform.
Proof
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
When this page/section has been completed, {{ProofWanted}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Source of Name
This entry was named for Ulisse Dini.
