1.5. Sup and Inf 27
Proof. Suppose A is a non-empty subset of R which is bounded below . We must
show that there is a lower bound for A which is greater than any other lower bound
for A. If m is any lower bound for A, then Example 1.3.8(a) implies that −m is an
upper bound for −A = {−a : a A}. Since R is a complete ordered field, there is
a least upper bound r for −A. Then
−a r for all a A and r −m.
Applying Example 1.3.8(a) yields that
−r a for all a A and m −r.
Thus, −r is a lower bound for A and, since m was an arbitrary lower bound, the
inequality m −r implies that −r is the greatest lower bound.
The Extended Real Numbers. For many reasons, it is convenient to extend
the real number system by adjoining two new points and −∞. The resulting
set is called the extended real number system. We declare that is greater than
every other extended real number and −∞ is less than every other extended real
number. This makes the extended real number system an ordered set. We also
define x + to be if x is any extended real number other than −∞. Similarly,
x = x + (−∞) is defined to be −∞ if x is any extended real number other
than ∞. Of course, there is no reasonable way to make sense of ∞.
The introduction of the extended real number system is just a convenient no-
tational convention. For example, it allows us to make the following definition.
Sup and Inf.
Definition 1.5.2. Let A be an arbitrary non-empy subset of R. We define the
supremum of A, denoted sup A, to be the smallest extended real number M such
that a M for every a A.
The infimum of A, denoted inf A, is the largest extended real number m such
that m a for all a A.
Note that, if A is bounded above, then sup A is the least upper bound of A. If
A is not bounded above, then the only extended real number M with a M for
all a A is ∞, and so sup A = in this case. Similarly, inf A is the greatest lower
bound of A if A is bounded below and is −∞ if A is not bounded below. Thus,
sup A and inf A exist as extended real numbers for any non-empty set A, but they
might not be finite. Also note that, even when they are finite real numbers, they
may not actually belong to A, as Example 1.4.6 shows.
Example 1.5.3. Find the sup and inf of the following sets:
A = (−1,1] = {x R : −1 x 1};
B = (−∞,5) = {x R : x 5};
C =
n + 1
: n N ; (1.5.1)
D =
: n N . (1.5.2)
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