MthT 430 Notes Chapter 8d Least Upper Bounds

chap8d.pdf

Least Upper Bounds

Recall the definitions of upper bound and least upper bound .

Definition. A set A of real numbers is bounded above if there is a number x such that

x ≥ a    for every a in A.

Such a number x is called an upper bound for A.

Definition. A number x is a least upper bound for a set A if

x is an upper bound for A, (1) if y is an upper bound for A, then x ≤ y. (2)

Such a number x is also called the supremum for A and sometimes denoted by supA or lubA.

There is an equivalent definition of least upper bound .

Definition. A number x is a least upper bound for a set A if

⎧ ⎪ ⎨ ⎪ ⎩ x is an upper bound for A, (1) For every ϵ > 0, there is an xϵ ∈ A such that x − ϵ < xϵ ≤ x. (2′)

Such a number x is also called the supremum for A and sometimes denoted by supA or lub A.

To show that the two definitions are equivalent, we must prove the following If and Only If Theorem :

Theorem. If x is an upper bound for A,then

If y is an upper bound for A, then x ≤ y. (2)

if and only if

For every ϵ > 0, there is an xϵ ∈ A such that x − ϵ < xϵ ≤ x. (2′)

Proof: First (2) ⇒ (2′). Assume (2). The proof is by contradiction. Assume there IS an ϵ > 0 such that there is no x_ϵ ∈ A such that x − ϵ < x_ϵ ≤ x. But then x − ϵ would be an upper bound for A which is less than x.

Second (2′) ⇒ (2). Again use contradiction. If (2) is false and (2) is true, there is an upper bound b for A which satisfies b < x. Let ϵ = x − b > 0. There is no x_ϵ ∈ A such that x − ϵ = b < x_ϵ ≤ x.

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