# proof of Dini’s theorem

Without loss of generality we will assume that $X$ is compact and, by replacing $f_{n}$ with $f-f_{n}$, that the net converges monotonically to 0.

Let $\epsilon>0$. For each $x\in X$, we can choose an $n_{x}$, such that $f_{n_{x}}(x)<\epsilon/2$. Since $f_{n_{x}}$ is continuous, there is an open neighbourhood $U_{x}$ of $x$, such that for each $y\in U_{x}$, we have $f_{n_{x}}(y)<\epsilon/2$. The open sets $U_{x}$ cover $X$, which is compact, so we can choose finitely many $x_{1},\ldots,x_{k}$ such that the $U_{x_{i}}$ also cover $X$. Then, if $N\geq n_{x_{1}},\ldots,n_{x_{k}}$, we have $f_{n}(x)<\epsilon$ for each $n\geq N$ and $x\in X$, since the sequence $f_{n}$ is monotonically decreasing. Thus, $\{f_{n}\}$ converges to 0 uniformly on $X$, which was to be proven.

Title proof of Dini’s theorem ProofOfDinisTheorem 2013-03-22 12:44:13 2013-03-22 12:44:13 mathcam (2727) mathcam (2727) 5 mathcam (2727) Proof msc 54A20