Introduction to Real Analysis

Dr. Weihu Hong

Clayton State University

8/21/2008

Upper Bound of a set Definition of upper bound of a set: A subset E of R is

bounded above if there exists β in R such that for each x in E, x β. Such a β is called an upper bound of E.

How should you define “bounded below” and “lower bound”? A set E is bounded if E is bounded both above and below. Examples:

A = {0, ½, 2/3, ¾, …}. Is A bounded below? Is A bounded above? Is A bounded?

N = {1, 2, 3, …}. Is N bounded below? Is N bounded above? Is N bounded?

B = {r belongs to Q: 0 < r and r²<2}. Is B bounded below? Is B bounded above? Is B bounded? Does B have the maximum element?

Least upper bound of a set

Definition of the least upper bound or supremum: Let E be a nonempty subset of R that is bounded above. An element α of R is called the least upper bound or supremum of E if (i) α is an upper bound of E, and (ii) if a real number β < α, then β is not an upper

bound of E. If E has the least upper bound, then the least

upper bound is denoted by α=sup E.

Greatest lower bound or infimum

Definition of the greatest lower bound or infimum: Let E be a nonempty subset of R that is bounded below. An element α of R is called the greatest lower bound or infimum of E if (i) α is a lower bound of E, and (ii) if a real number β > α, then β is not a lower

bound of E. If E has the greatest lower bound, then the

greatest lower bound is denoted by α=inf E.

Theorem 1.4.4

Let A be a nonempty subset of R that is bounded above. An upper bound α of A is the supremum of A if and only if for every β < α, there exists an element x of A such that β<x α.

Proof: “=>”: Suppose α=sup A. If β<α, then β is not an upper

bound of A. Thus there exists an element x in A such that x > β. Since α is an upper bound of A, therefore, x α.

“<=”: if α is an upper bound of A such that every β < α is not an upper bound of A, then it follows from the definition that α is the least upper bound of A, that is, α = sup A.

New definition of the least upper bound of a set

It follows from the theorem 1.4.4 that we might define the least upper bound of a set as follows:

Let E be a nonempty subset of R that is bounded above. An element α of R is called the least upper bound or supremum of E if (i) α is an upper bound of E, and (ii) for every ε>0, there exists an element x of E such that

α - ε <x α.

Examples

A = {0, ½, 2/3, ¾, …}. Find the inf A and sup A.

N = {1, 2, 3, …}. Find the inf N and sup N.

B = {r belongs to Q: 0 < r and r²<2}. Find inf B and sup B.

Least Upper Bound Property of R (Cantor and Dedekind)

Property 1.4.6: Supremum or Least Upper Bound Property of R: Every nonempty subset of R that is bounded above has a supremum in R.

Property 1.4.7: Infimum or Greatest Lower Bound Property of R: Every nonempty subset of R that is bounded below has an infimum in R.

Example

Show that for every positive real number y>1, there exists a unique positive square root of y, that is,

Proof: Existence: Consider the set

Is S nonempty? Is S bounded above?

Let α=sup S. Show α² = y.

y

}0:{ 2 yxandxRxS

Use contradiction to show α²=y Define the real number β by

Then

If α²<y, then by (1), β>α, and by (2), β²<y. This contradicts that α is an upper bound for S. On the other hand, if α²>y, then by (1), β<α and by (2), β²>y. Thus if x is a positive real number and x≥β,then x²≥ β²>y. Therefore, β is an upper bound of S. This contradicts that α is the least upper bound of S. Thus it must be true that α²=y.

)1()1(2

y

y

y

y

)2()(

))(1(2

22

y

yyyy

New definitions

If E is a nonempty subset of R and E is not bounded above, we set sup E =∞

If E is a nonempty subset of R and E is not bounded below, we set inf E = -∞