another proof of Dini’s theorem

This is the version of the Dini’s theoremMathworldPlanetmath I will prove: Let K be a compactPlanetmathPlanetmath metric space and (fn)nNC(K) which convergesPlanetmathPlanetmath pointwise to fC(K).

Besides, fn(x)fn+1(x)xK,n.

Then (fn)nN converges uniformly in K.


Suppose that the sequence does not converge uniformly. Then, by definition,

ε>0such that mNnm>m,xmKsuch that |fnm(xm)-f(xm)|ε.


For m=1n1>1,x1Ksuch that |fn1(x1)-f(x1)|εn2>n1,x2Ksuch that |fn2(x2)-f(x2)|εnm>nm-1,xmKsuch that |fnm(xm)-f(xm)|ε

Then we have a sequence (xm)mK and (fnm)m(fn)n is a subsequence of the original sequence of functions. K is compact, so there is a subsequence of (xm)m which converges in K, that is, (xmj)j such that


I will prove that f is not continuousPlanetmathPlanetmath in x (A contradictionMathworldPlanetmathPlanetmath with one of the hypothesisMathworldPlanetmathPlanetmath).

To do this, I will show that f(xmj)j does not converge to f(x), using above’s ε.

Let j0such that jj0|fnmj(x)-f(x)|<ε/4, which exists due to the punctual convergence of the sequence. Then, particularly, |fnmjo(x)-f(x)|<ε/4.

Note that


because (using the hypothesis fn(y)fn+1(y)yK,n) it’s easy to see that


Then, fnmj(xmj)-f(xmj)εj. And also the hypothesis implies


So, jj0fnmj0(xmj)fnmj(xmj), which implies




and so


On the other hand,


And as fnmj0 is continuous, there is a j1 such that




which implies


Then, particularly, f(xmj)j does not converge to f(x). QED.

Title another proof of Dini’s theorem
Canonical name AnotherProofOfDinisTheorem
Date of creation 2013-03-22 14:04:37
Last modified on 2013-03-22 14:04:37
Owner gumau (3545)
Last modified by gumau (3545)
Numerical id 11
Author gumau (3545)
Entry type Proof
Classification msc 54A20