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
We start with a straightforward definition similar to many others
in this course. Read the definitions carefully, and note the
If we are interested in the
best upper bound
best lower bound we need to consider the following.
We say that
The problem with maximums and minimums is that they are only
guaranteed to exist for finite sets .
For example, =
Completeness of reals, supremum form.
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 (
Then there is some
This completes the proof.
Completeness of reals, infimum form.
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