1.5. Sup and Inf 31
The following theorem concerning sup and inf for functions follows easily from
Theorem 1.5.7. We leave the details to the exercises.
Theorem 1.5.10. Let f and g be functions defined on a set containing A as a
subset, and let c R be a positive constant. Then
(a) supA cf = c supA f and infA cf = c infA f;
(b) supA(−f) = infA f;
(c) supA(f + g) supA f + supA g and infA f + infA g infA(f + g);
(d) sup{f(x) f(y) : x, y A} = supA f infA f.
Exercise Set 1.5
1. For each of the following sets, find the set of all extended real numbers x that
are greater than or equal to every element of the set. Then find the sup of the
set. Does the set have a maximum?
(a) (−10,10).
(b) {n2 : n N}.
2n + 1
n + 1
2. Find the sup and inf of the following sets. Tell whether each set has a maximum
or a minimum.
(a) (−2,8].
n + 2
n2 + 1
(c) {n/m : n, m Z,
n2 5m2}.
3. Prove that if sup A ∞, then for each n N there is an element an A such
that sup A 1/n an sup A.
4. Prove that if sup A = ∞, then for each n N there is an element an A such
that an n.
5. Formulate and prove the analog of Theorem 1.5.4 for inf.
6. Prove part (d) of Theorem 1.5.7.
7. Prove part (e) of Theorem 1.5.7.
8. If A and B are two non-empty sets of real numbers, then prove that
sup(A B) = max{sup A, sup B} and inf(A B) = min{inf A, inf B}.
9. Find supI f and infI f for the following functions f and sets I. Which of these
is actually the maximum or the minimum of the function f on I?
(a) f(x) =
I = [−1,1].
(b) f(x) =
x + 1
x 1
, I = (1,2).
(c) f(x) = 2x x2, I = [0,1).
10. Prove (a) of Theorem 1.5.10.
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