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}.

(c)

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].

(b)

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) =

x2,

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.