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 =

n2

n + 1

: n ∈ N ; (1.5.1)

D =

1

n

: n ∈ N . (1.5.2)