Proof of Baroni’s theorem

Let m=infA and M=supA . If m=M we are done since the sequenceMathworldPlanetmath is convergentMathworldPlanetmath and A is the degenerate interval composed of the point l¯ , where l=limnxn.

Now , assume that m<M . For every λ(m,M) , we will construct inductively two subsequencesMathworldPlanetmath xkn and xln such that limnxkn=limnxln=λ and xkn<λ<xln

From the definition of M there is an N1 such that :


Consider the set of all such values N1 . It is bounded from below (because it consists only of natural numbersMathworldPlanetmath and has at least one element) and thus it has a smallest element . Let n1 be the smallest such element and from its definition we have xn1-1λ<xn1 . So , choose k1=n1-1 , l1=n1 . Now, there is an N2>k1 such that :


Consider the set of all such values N2 . It is bounded from below and it has a smallest element n2 . Choose k2=n2-1 and l2=n2 . Now , proceed by inductionMathworldPlanetmath to construct the sequences kn and ln in the same fashion . Since ln-kn=1 we have :


and thus they are both equal to λ.

Title Proof of Baroni’s theoremMathworldPlanetmath
Canonical name ProofOfBaronisTheorem
Date of creation 2013-03-22 13:32:33
Last modified on 2013-03-22 13:32:33
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 5
Author mathwizard (128)
Entry type Proof
Classification msc 40A05