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
Canonical name	ProofOfDinisTheorem
Date of creation	2013-03-22 12:44:13
Last modified on	2013-03-22 12:44:13
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	5
Author	mathcam (2727)
Entry type	Proof
Classification	msc 54A20