1549901369-Elements_of_Real_Analysis__Denlinger_

358 Chapter 7 • The Riemann Integral

Theorem 7.1.2 Suppose A~ B ~JR, where B is bounded. Then

(a) sup A ::::; sup B, and

(b) inf A .;::: inf B.

Proof. Exercise l. •

Theorem 7.1.3 If A~ JR is bounded and x E JR, then

(a) sup(x +A) = x +sup A, and inf(x +A) = x +inf A

(b) If x > 0, then sup(xA) = x sup A, and inf(xA) = x inf A;

(c) sup(-A) = - inf A, and inf(-A) = - sup A;

(d) If x < 0, then sup(xA) = x inf A, and inf(xA) = x sup A.

Proof. Suppose A ~ JR, x E JR, and u = sup A E R

(a) Let y Ex+ A. Then y = x +a for some a EA. But x +a::::; x + u, so

y ::::; x + u. Thus, x + u is an upper bound for x + A.

Suppose v is another upper bound for x +A. Then Va E A, x +a ::::; v.

Thus, Va E A, a ::::; v - x. So, v - x is an upper bound for A. Hence, u ::::; v - x.

Therefore,

x + u::::; v.

Putting the above results together with Definition 1.6.3, x+u = sup(x+A).

That is, x+supA = sup(x+A). The proof that inf(x+A) = x+inf A is similar.

Proof of (b)-(d): Exercise 2. •

Theorem 7.1.4 For A, B ~JR, define A+ B ={a+ b : a EA, b EB}.

(a) If A and B are bounded below, then inf(A + B) =inf A+ inf B.

(b) If A and B are bounded above, then sup( A + B) = sup A + sup B.

Proof. Exercise 3. •

Theorem 7.1.5 If A, B are nonempty sets of real numbers such that Va EA,
Vb E B, a ::::; b, then sup A ::::; inf B, and the following are equivalent:

(a) sup A= inf B.

(b) Vc>O, :la EA, bEB3b-a<c.

(c) :JK > 0 3 Ve> 0, :la EA, b EB 3 b-a< Kc.

( d) :J one and only one real number u 3 Va E A , b E B, a ::::; u ::::; b.

(In this case, u = sup A = inf B.)