Dini's Theorem
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This theorem requires a proof..
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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
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Source of Name
This entry was named for Ulisse Dini.