nested interval theorem

Proposition 1.



is a sequence of nested closed intervalsMathworldPlanetmathPlanetmath, then


If also  limn(bn-an)=0,  then the infinite intersection consists of a unique real number.


There are two consequences to nesting of intervals: [am,bm][an,bn] for nm:

  1. 1.

    first of all, we have the inequalityMathworldPlanetmath anam for nm, which means that the sequence a1,a2,,an, is nondecreasing;

  2. 2.

    in addition, we also have two inequalities: ambn and anbm. In either case, we have that aibj for all i,j. This means that the sequence a1,a2,,an, is bounded from above by all bi, where i=1,2,.

Therefore, the limit of the sequence (ai) exists, and is just the supremum, say a (see proof here ( Similarly the sequence (bi) is nonincreasing and bounded from below by all ai, where i=1,2,, and hence has an infimum b.

Now, as the supremum of (ai), abi for all i. But because b is the infimum of (bi), ab. Therefore, the interval [a,b] is non-empty (containing at least one of a,b). Since aiabbi, every interval [ai,bi] contains the interval [a,b], so their intersection also contains [a,b], hence is non-empty.

If c is a point outside of [a,b], say c<a, then there is some ai, such that c<ai (by the definition of the supremum a), and hence c[ai,bi]. This shows that the intersection actually coincides with [a,b].

Now, since limn(bn-an)=0, we have that b-a=limnbn-limnan=limn(bn-an)=0. So a=b. This means that the intersection of the nested intervals contains a single point a. ∎

Remark.  This result is called the nested interval theorem. It is a restatement of the finite intersection property for the compact set[a1,b1].  The result may also be proven by elementary methods: namely, any number lying in between the supremum of all the an and the infimum of all the bn will be in all the nested intervals.

Title nested interval theorem
Canonical name NestedIntervalTheorem
Date of creation 2013-03-22 17:27:12
Last modified on 2013-03-22 17:27:12
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Theorem
Classification msc 54C30
Classification msc 26-00