Dini's Theorem

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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.


Source of Name

This entry was named for Ulisse Dini.