Dini's theorem

Proof

Let ${\displaystyle \epsilon >0}$ be given. For each ${\displaystyle n\in \mathbb{N}}$, let ${\displaystyle g_n=f-f_n}$, and let ${\displaystyle E_n}$ be the set of those ${\displaystyle x\in X}$ such that ${\displaystyle g_n(x)<\epsilon }$. Each ${\displaystyle g_n}$ is continuous, and so each ${\displaystyle E_n}$ is open (because each ${\displaystyle E_n}$ is the preimage of the open set ${\displaystyle (-\mathrm{\infty },\epsilon )}$ under ${\displaystyle g_n}$, a continuous function). Since ${\displaystyle (f_n{)}_{n\in \mathbb{N}}}$ is monotonically increasing, ${\displaystyle (g_n{)}_{n\in \mathbb{N}}}$ is monotonically decreasing, it follows that the sequence ${\displaystyle E_n}$ is ascending (i.e. ${\displaystyle E_n\subset E_{n+1}}$ for all ${\displaystyle n\in \mathbb{N}}$). Since ${\displaystyle (f_n{)}_{n\in \mathbb{N}}}$ converges pointwise to ${\displaystyle f}$, it follows that the collection ${\displaystyle (E_n{)}_{n\in \mathbb{N}}}$ is an open cover of ${\displaystyle X}$. By compactness, there is a finite subcover, and since ${\displaystyle E_n}$ are ascending the largest of these is a cover too. Thus we obtain that there is some positive integer ${\displaystyle N}$ such that ${\displaystyle E_N=X}$. That is, if ${\displaystyle n>N}$ and ${\displaystyle x}$ is a point in ${\displaystyle X}$, then ${\displaystyle |f(x)-f_n(x)|<\epsilon }$, as desired.