30 1. The Real Numbers
(c) Since a sup A and b sup B for all a A, b B, we have
a + b sup A + sup B for all a A, b B.
It follows that
sup(A + B) sup A + sup B.
Let x be any number less than sup A+sup B. We claim that there are elements
a A and b B such that
(1.5.3) x a + b.
Once proved, this will imply that no number less than sup A + sup B is an upper
bound for A + B. Thus, proving this claim will establish that sup(A + B) =
sup A + sup B.
There are two cases to consider: sup B finite and sup B = ∞. If sup B is
finite, then x sup B sup A, and Theorem 1.5.4 implies there is an a A with
x sup B a. Then x a sup B. Applying Theorem 1.5.4 again, we conclude
there is a b B with x a b. This implies (1.5.3) and proves our claim in the
case where sup B is finite.
Now suppose sup B = ∞. Let a be any element of A. Then x−a sup B =
and so, as above, we conclude from Theorem 1.5.4 that there is a b B satisfying
x−a b. This implies (1.5.3), which establishes our claim in this case and completes
the proof.
Sup and Inf for Functions. If f is a real-valued function defined on some set
X and if A is a subset of X, then
f(A) = {f(x) : x A}
is a set of real numbers, and so we can take its sup and inf.
Definition 1.5.8. If f : X R is a function and A X, then we set
sup
A
f = sup{f(x) : x A} and inf
A
f = inf{f(x) : x A}.
Thus, supA f is the supremum of the set of values that f assumes on A and
infA f is the infimum of this set. They themselves may or may not be values that
f assumes on A. If supA f is a value that f assumes on A, then it is called the
maximum of f on A. Similarly, if infA f is a value assumed by f somewhere on A,
then it is called the minimum of f on A.
Example 1.5.9. Find supI f and infI f if
(a) f(x) = sin x and I = [−π/2,π/2);
(b) f(x) = 1/x and I = (0, ∞).
Solution: (a) The function sin x takes on all values in the interval [−1,1) on
I but does not take on the value 1. Thus, infI f = −1 and supI f = 1. In this case,
infI f is a value assumed by f on I, but supI f is not.
(b) The function 1/x takes on all values in the open interval (0, ∞). Thus,
infI f = 0 and supi f = in this case. Neither one of these extended real numbers
is a value taken on by f on I.
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