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

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

Definition.

If we are interested in the best

upper bound
or best

lower bound we need to consider the following.

Definition.

We say that **
** is a least upper bound for
if is an upper bound for and no

The problem with maximums and minimums is that they are only
*guaranteed* to exist for *finite* sets .
For example, =**=
**
has both a maximum (2) and a minumum (0).

In fact, for a nonempty set , the maximum element of
exists if and only if

Let =

Let =

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 =

Let =

Completeness of reals, supremum form.

**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 _{0
} with some
_{0
}. This just
uses the assumption that is nonempty. Now
inductively assume that _{
} is defined and
_{
} for some _{
} is not an upper bound for .
If _{
+1
} is an upper bound for
then let _{
+1=}_{
}.
Otherwise, let _{
+1=}_{
+
}
where _{0-1
}
_{0}
_{
+
0
} for all
**
** and we need only
take

By construction, (_{
)
} is a nondecreasing sequence
of rationals and bounded above by , an upper bound for
. It follows that (_{
)
} converges to
some

**Subproof.**

**Subproof.**

Then there is _{
+1=}_{
+
}
with _{
+
}. But
_{
+
-1
} so
_{
+
+1
+1
}
contradicting this choice of

Thus

**Subproof.**

This completes the proof.

Completeness of reals, infimum form.

**Proof.**

Let **= -**. Then

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