alternate statement of Bolzano-Weierstrass theorem


Let S be bounded and infinite. Since S is bounded there exist a,b, with a<b, such that S[a,b]. Let b-a=l and denote the midpointMathworldPlanetmathPlanetmathPlanetmath of the intervalMathworldPlanetmathPlanetmath [a,b] by m. Note that at least one of [a,m],[m,b] must contain infinitely many points of S; select an interval satisfying this condition, denoting its left endpoint by a1 and its right endpoint by b1. Continuing this process inductively, for each n, we have an interval [an,bn] satisfying

[an,bn][an-1,bn-1][a1,b1][a,b], (1)

where, for each i such that 1in, the interval [ai,bi] contains infinitely many points of S and is of length l/2i. Next we note that the set A={a1,a2,an} is contained in [a,b], hence is bounded, and as such, has a supremum which we denote by x. Now, given ϵ>0, there exists N such that x-ϵ<aNx. Furthermore, for every mN, we have x-ϵ<aNamx. In particular, if we select mN such that l/2m<ϵ, then we have

x-ϵ<anamxbm=am+l2m<x+ϵ. (2)

Since [am,bm](x-ϵ,x+ϵ) contains infinitely many points of S, we may conclude that x is a limit point of S. ∎

Title alternate statement of Bolzano-Weierstrass theorem
Canonical name AlternateStatementOfBolzanoWeierstrassTheorem
Date of creation 2013-03-22 16:40:13
Last modified on 2013-03-22 16:40:13
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Theorem
Classification msc 26A06
Related topic BolzanoWeierstrassTheorem
Related topic LimitPoint
Related topic Bounded
Related topic Infinite