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 :
nor do these axioms even imply that such sequences have limits. (In fact 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
We have already given the definition of a monotonic sequence. Because of its importance for the material being considered, we shall repeat it here.
A sequence is monotonic nondecreasing if for all . It is monotonic nonincreasing if for all . It is monotonic if it is either monotonic nondecreasing or monotonic nonincreasing.
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
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.
An Archimedean ordered field is said to be complete if whenever is a sequence of numbers from and is bounded and monotonic nondecreasing then there is such that as .
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
will be able to derive the next theorem.
Monotone Convergence Theorem.
Let be a complete ordered field such as . Let be a sequence of numbers from such that is bounded and monotonic nondecreasing. Then there is such that as .
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
every Cauchy sequence converges to a limit,
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.
Let be a complete ordered field such as . Let be a sequence of numbers from such that is bounded and monotonic nonincreasing. Then there is such that as .
Consider the sequence 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 is a nondecreasing sequence of real numbers, then is bounded below by its first term, . So it is only necessary to show that is bounded above to conclude that has a limit. Suppose we can do this. Then there is such that for all , say. It follows by the theorem giving bounds on limits that the limit of must satisfy . Also, by results given earlier, we have for all .
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.
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.