The Monotone Convergence Theorem and Completeness of the Reals

1. Introduction

The axioms for Archimedean ordered fields allow us to define and describe sequences and their limits and prove many results about them, but do not distinguish the field of rationals from the field of real numbers and do not explain the convergence of familiar sequences such sequence from the decimal expansion of π :

3 , 3.1 , 3.14 , 3.141 , 3.1415 , 3.14159 ,

nor do these axioms even imply that such sequences have limits. (In fact π is not in , so the Archimedean ordered field does not contain the limit of this sequence.)

We need to add a further axiom for the reals, called completeness that will enable us to conclude that sequences like the one above have limits. This web page discusses these points. A further web page will present some nice examples.

2. Monotonic sequences and the completeness axiom

We have already given the definition of a monotonic sequence. Because of its importance for the material being considered, we shall repeat it here.

Definition.

A sequence ( a n ) is monotonic nondecreasing if a n + 1 a n for all n . It is monotonic nonincreasing if a n + 1 a n for all n . It is monotonic if it is either monotonic nondecreasing or monotonic nonincreasing.

By thought experiments or real-world experience (such as those given in the lectures of this course) we believe that any bounded monotonic sequence should have a limit. Such arguments are not real proofs in mathematics, but nevertheless very much part of mathematics. The right way to use such informal arguments is as follows: first make your argument as precisely as possible to justify a property or properties of the system we are interested in (remember, it is informal so cannot be a completely watertight proof); second, add the additional properties that argument attempts to justify as axioms, so that at all stages it will be 100% clear what assumptions are being made about this system in our formal theory and rigorous proofs.

It may be that someone coming along later may not accept our informal arguments but may like our axioms for different reasons. This person will then be able to see the assumptions clearly and quickly and accept the proofs from them as valid and useful.

In the case of the reals, and our thought experiments on them, we are lead to state formally a new axiom for the reals, an additional axiom that may or may not be true for an Archimedean ordered field, but which we believe is true for the reals.

Completeness Axiom.

An Archimedean ordered field F is said to be complete if whenever ( a n ) is a sequence of numbers from F and ( a n ) is bounded and monotonic nondecreasing then there is l F such that a n l as n .

A complete Archimedean ordered field will be called a complete ordered field for brevity.

The reals, , forms an example of a complete ordered field. In fact in many ways, it is the only example, a point that may be discussed further later.

Everyone, however good or bad at writing proofs, will be able to derive the next theorem.

Monotone Convergence Theorem.

Let F be a complete ordered field such as . Let ( a n ) be a sequence of numbers from F such that ( a n ) is bounded and monotonic nondecreasing. Then there is l F such that a n l as n .

Proof.

A triviality, as the conclusion is the same as one of our axioms.

What is happening here is that there are several different formulations of the completeness axiom. One of these is the Monotone Convergence Theorem itself. Two others are the statements that every Cauchy sequence converges to a limit, and every nonempty bounded set has a least upper bound, both of which will be discussed later. I have taken one particular version of the completeness axiom, and this one makes the proof of the Monotone Convergence Theorem a triviality. But later on I will have to prove that from this particular completeness axiom that every Cauchy sequence converges to a limit, and that every nonempty bounded set has a least upper bound. Someone who prefers a different version of completeness will have to prove the Monotone Convergence Theorem from it. Hopefully, in the end everyone will agree that these three assertions are all true for the reals.

There seems to be a law in mathematics that you have to do the same amount of work eventually, whatever approach you take to a problem. This seems to be the case here, and we will have to do some more work later. Actually, there's another law that says if you approach a problem in the right way you can often simplify the problem or reduce the amount of work considerably. This law also applies here, as the monotone convergence theorem is (I think!) a much more convincing axiom in that it is obviously true for the reals than the other two.

Of course, Good Mathematicians know when to apply the first law and when to apply the second law. Hopefully you are (or will soon be) a Good Mathematician.

The next result is not a complete triviality in the same sense, but is very very easy.

Theorem.

Let F be a complete ordered field such as . Let ( a n ) be a sequence of numbers from F such that ( a n ) is bounded and monotonic nonincreasing. Then there is l F such that a n l as n .

Proof.

Consider the sequence ( - a n ) and apply the Monotone Convergence Theorem to it.

There are a few useful remarks concerning monotonic sequences that will help you get a picture of how they work. The results on monotonic sequences from a previous web page will also help. The main point is that if ( a n ) is a nondecreasing sequence of real numbers, then ( a n ) is bounded below by its first term, a 1 . So it is only necessary to show that ( a n ) is bounded above to conclude that ( a n ) has a limit. Suppose we can do this. Then there is B such that a n B for all n , say. It follows by the theorem giving bounds on limits that the limit l = lim n a n of a n must satisfy l B . Also, by results given earlier, we have a k l for all k .

Similar remarks to those in the last paragraph apply to nonincreasing sequences. I'll leave you to flip the order over in your head and work out what they should be.

In fact, there is a very beautiful result that says that any two complete ordered fields are isomorphic. This is, to my mind, one of the nicest sort of result that says that we have specified our real number line exactly.

The way I like to think of this is by saying if some aliens were to visit us on Earth and want to have a conversation with us about real numbers, we would be able to have a productive time. We would start by explaining axioms to each other. Most likely the axioms we would choose would be different, but since the axioms I have presented characterise the real numbers exactly we should be able to prove ET's axioms from ours (and ET would be able to prove ours from his/hers/its axioms) and prove the two systems equivalent. Then would know that, whatever our informal conception or picture of the reals, we are actually talking about the same mathematical object. Further discussion of this point for interested students (not examinable) is on a separate web page on the uniqueness of the reals after a number of rather different constructions of the reals have been presented.

3. Conclusion

You have seen the completeness axiom for the reals, in the rather useful form of the Monotone Convergence Theorem. It turns out that any complete Archimedean ordered field is isomorphic to the reals, so these axioms capture the properties of the reals exactly.