Dini’s_Theorem

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Definition

Assume $f_n \to f$ pointwise on a compact set $K$ and assume that for each $x \in K$ the sequence $(f_n(x))$ is increasing. If $f_n$ and $f$ are all continuous on $K$, then the convergence is uniform.

Proof

Let $g_n = f - f_n$. Then $g_n \geq 0$, $g_n$ is continuous, $g_n \to 0$ pointwise, and $(g_n(x))$ is decreasing for each $x$.

Given $\epsilon > 0$, define $K_n = \{x \in K : g_n(x) \geq \epsilon\}$. Each $K_n$ is closed (by continuity) and $K_n \supseteq K_{n+1}$ (since $g_n$ is decreasing).

Since $g_n \to 0$ pointwise, $\bigcap K_n = \emptyset$. By compactness, some $K_N = \emptyset$.

Thus for $n \geq N$, $g_n(x) < \epsilon$ for all $x \in K$, giving uniform convergence.