Supremum and infimum

1. Introduction

We have already seen two equivalent forms of the completeness axiom for the reals: the montone convergence theorem and the statement that every Cauchy sequence has a limit. The second of these is useful as it doesn't mention the order relation and so applies to the complex numbers for instance. (It turns out that the set of complex numbers is also complete in the Cauchy sense.) This web page describes a third approach due to Dedekind. This uses the order relation of but applies to arbitrary subsets of rather than sequences. There are a number of places (including results about the radius of convergence for power series, and several results in more advanced analysis) where Dedekind's approach is particularly helpful.

2. Supremum and infimum

We start with a straightforward definition similar to many others in this course. Read the definitions carefully, and note the use of and here rather than and .

Definition.

Let .

We say that is bounded above if there is such that ( ) . The number is called an upper bound for .

We say that is bounded below if there is such that ( ) . The number is called an lower bound for .

If we are interested in the best upper bound or best lower bound we need to consider the following.

Definition.

Let .

We say that is a least upper bound for if is an upper bound for and no is also an upper bound. In other words if ( ) and ( ) . The least upper bound of a set may not exist, but if it does it is unique, because if we have two distince upper bounds , then one of these must be larger and so it cannot be a least upper bound. When it exists, the least upper bound of a set is called the supremum of and denoted sup .

We say that is a greatest lower bound for if is a lower bound for and no is also a lower bound. In other words if ( ) and ( ) . The greatest lower bound of a set may not exist, but if it does it is unique, and is called the infimum of and denoted inf .

Supremums and infimums are a bit like maximums and minimums for infinite sets .

The problem with maximums and minimums is that they are only guaranteed to exist for finite sets . For example, = 0 2 doesn't have a maximum element, though it does have a minimum element, 0. On the other hand = 0 4 has both a maximum (2) and a minumum (0).

In fact, for a nonempty set , the maximum element of exists if and only if sup exists and is an element of . Similarly, the minimum element of exists if and only if inf exists and is an element of .

Let =1-1 . Then has no largest element, i.e., max doesn't exist, but sup=1 since 1 is an upper bound and any 1 is less than some 1-1 by the Archimedean property. Note that sup is not an element of .

Let = 1 . Then has no smallest element, i.e., min doesn't exist, but inf=0 since 0 is a lower bound and any 0 is greater than some 1 by the Archimedean property. Note that inf is not an element of .

Let =2,3 . In this case does have largest and smallest elements, and sup=3 and inf=2 .

Let be the empty set. Then by convention every is both an upper bound and a lower bound for . So does not have least upper bound or greatest lower bound.

Let = . Then does not have any upper bound, by the Archimedean property. But does have a lower bound, such as -1. The greatest lower bound is determined by your convention on the set of natural numbers. If you prefer 0 then inf=0 . Otherwise you will have inf=1 .

Let = . Then does not have any upper bound nor any lower bound, by the Archimedean property again.

Completeness of reals, supremum form.

Let be non-empty and bounded above. Then there is a least upper bound of , sup .

Proof.

This is proved by a variation of a proof of an earlier result that every real number has a monotonic sequence of rationals converging to it.

We start by defining a sequence (₎ using induction on . Start with any ₀ with some such that ₀. This just uses the assumption that is nonempty. Now inductively assume that is defined and for some , i.e., is not an upper bound for . If ₊₁ is an upper bound for then let ₊₁₌. Otherwise, let ₊₁₌₊ where = _0-1 and ₀ is chosen to be the least natural number such that _+

₀ for all . Such a exists since is bounded above by some and we need only take so that ₊, using the Archimedean Property. So since there is some such there must be a least such number, ₀ .

By construction, (₎ is a nondecreasing sequence of rationals and bounded above by , an upper bound for . It follows that (₎ converges to some . We shall show that this is the least upper bound of .

Subproof.

Suppose is not an upper bound of .

Subproof.

Then there is such that . But this gives a contradiction, for if is such that - 1 we consider the th stage of the construction, where we chose ₊₁₌₊ with greatest so that there is some with ₊. But _+

-1 so _+
+1

+1 contradicting this choice of .

Thus is after all an upper bound. To see it is the least upper bound, we again suppose otherwise.

Subproof.

Suppose is not the least upper bound of .

Subproof.

Then there is some such that every has . This again is impossible. Since , there must be some with . (To see this, put =- and with _- .) But by construction of there is always some with . Therefore and is not after all an upper bound for .

This completes the proof.

Completeness of reals, infimum form.

Let be non-empty and bounded below. Then there is a greatest lower bound of , inf .

Proof.

Let be a lower bound for and let =- . Then =- is an upper bound for and hence by the previous result has a least upper bound, . It follows easily that - is a greatest lower bound for , for if then - as - so - , and if - then - so - is not an upper bound for , so there is with - hence - and clearly - .

These results are equivalent to the monotone convergence theorem. To see this, suppose (₎ is a bounded nondecreasing sequence. Then let = . By the fact that the sequence is bounded and by completeness theorem above, =sup exists, and it is a nice exercise to show that as .

3. Summary

You have seen another form of the completeness axiom for the reals, which is useful in that it doesn't involve any sequences and may be applied to subsets of .