Dini's Theorem

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.