CHAPTER
SEQUENCES;
INDETERMINATE FORMS;
IMPROPER INTEGRALS
10
±
10.1
THE LEAST UPPER BOUND AXIOM
So far our approach to the real number system has been somewhat primitive. We have
simply taken the point of view that there is a one-to-one correspondence between the
set of points on a line and the set of real numbers, and that this enables us to measure all
distances, take all roots of nonnegative numbers, and, in short, Fll in all the gaps left
by the set of rational numbers. This point of view is basically correct and has served
us well, but it is not sufFciently sharp to put our theorems on a sound basis, nor is it
sufFciently sharp for the work that lies ahead.
We begin with a nonempty set
S
of real numbers. As indicated in Section 1.2, a
number
M
is an
upper bound
for
S
if
x
≤
M
for all
x
∈
S
.
It follows that if
M
is an upper bound for
S
, then every number in [
M
,
∞
) is also an
upper bound for
S
. Of course, not all sets of real numbers have upper bounds. Those
that do are said to be
bounded above
.
It is clear that every set that has a largest element has an upper bound: if
b
is the
largest element of
S
, then
x
≤
b
for all
x
∈
S
. This makes
b
an upper bound for
S
. The
converse is false: the sets
S
1
=
(
−∞
, 0)
and
S
2
=
±
1
2
,
2
3
,
3
4
,
...
,
n
n
+
1
,
...
²
both have upper bounds (for instance, 2 is an upper bound for each set), but neither has
a largest element.
Let’s return to the Frst set,
S
1
. While (
−∞
, 0) does not have a largest element, the
set of its upper bounds, namely [0,
∞
), does have a smallest element, namely 0. We
call 0 the
least upper bound
of (
−∞
, 0).
Now let’s reexamine
S
2
. While the set of quotients
n
n
+
1
=
1
−
1
n
+
1
,
n
=
1, 2, 3,
...
,
585