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.

Proof
Negating both $f_n$ and $f$, if necessary, we may assume that:
 * $\forall x \in K : \sequence {\map {f_n} x}$ is monotonically decreasing

Define the positive functions by:
 * $g_n := f_n - f$

By construction:
 * $(G1)$ $g_n \ge 0$
 * $(G2)$ $g_n$ is continuous in view of the difference rule
 * $(G3)$ $\sequence {g_n}$ converge pointwise to $0$
 * $(G4)$ $\forall x \in K : \sequence {\map {g_n} x}$ is monotonically decreasing

Now we need to show that $\sequence {g_n}$ converges uniformly to $0$.

Let $\epsilon > 0$.

Consider $\sequence {A_n}$ defined by:
 * $A_n := \set {x \in K : \map {g_n} x < \epsilon }$

Then, $\sequence {A_n}$ is an increasing sequence of open sets, in view of $(G4)$ and $(G2)$.

Moreover, by $(G3)$, we have:
 * $K = \bigcup A_n$

Since $K$ is compact, there exists a finite subcover:
 * $K = A_{n_1} \cup \cdots \cup A_{n_m}$

for some $m \in \N_{>0}$.

Let $N := \max \set {n_1, \ldots ,n_m}$ be the maximum element among $n_1, \ldots, n_m$.

Then, since $\sequence {A_n}$ is increasing, we have:
 * $\forall n \ge N : K = A_n$

Together with $(G1)$, we have:
 * $\forall n \ge N : \forall x \in K : 0 \le \map {g_n} x \le \epsilon $