proof of Dini’s theorem

Without loss of generality we will assume that X is compactPlanetmathPlanetmath and, by replacing fn with f-fn, that the net convergesPlanetmathPlanetmath monotonically to 0.

Let ϵ>0. For each xX, we can choose an nx, such that fnx(x)<ϵ/2. Since fnx is continuousPlanetmathPlanetmath, there is an open neighbourhood Ux of x, such that for each yUx, we have fnx(y)<ϵ/2. The open sets Ux cover X, which is compact, so we can choose finitely many x1,,xk such that the Uxi also cover X. Then, if Nnx1,,nxk, we have fn(x)<ϵ for each nN and xX, since the sequence fn is monotonically decreasing. Thus, {fn} 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