below if it has a lower bound. Example 1. A set Acan have many upper bounds, but only one least upper bound. Let S ⊆ R be a set that is bounded above. A number B is called the least upper bound (or supremum) of the set S if: 1) B is an upper bound: any x ∈ S satisfies x ≤ B, and }\) The least-upper-bound property is sometimes called the completeness property or the Dedekind completeness property 1 . An ordered set \(S\) has the least-upper-bound property if every nonempty subset \(E \subset S\) that is bounded above has a least upper bound, that is \(\sup\, E\) exists in \(S\text{. The number M is called an upper bound for the set S. Note that if M is an upper bound for S then any bigger number is also an upper bound. The same is true of A = Q. A number u ∈ R is called the supremum (least upper bound) of S, denoted by supS, if it satisfies the conditions 1. s ≤ u for all s ∈ S . Least upper bound (LUB) refers to a number that serves as the lowest possible ceiling for a set of numbers. 2 Definition 2.2. 2. If a set of numbers has a greatest number, then that number is also the least upper bound (supremum). 10.1: Least upper bounds and greatest lower bounds. [Picture drawn in class.] contradiction, so smust be an upper bound for A. A maximum is always a least upper bound. For example, the set ofnatural numbers does not. Least Upper Bound of a Bounded Function. If M ∈ R is an upper bound of A such that M ≤ M′ for every upper bound M′ of A, then M is called the supremum of A, denoted M = supA. This theorem is known as the Archimedean property of real numbers. Definition 4. That is, an upper bound of S is a number α … We know that s 1 n is not an upper bound for A, so there exists an a2A such that a>s 1 n >s , and therefore sis the least upper bound for A. Let cdenote the least upper bound of the left endpoints and dthe greatest lower bound of the right endpoints. The argument for uniqueness is to assume that there are two least upper bounds s 1 and s 2 of a set A. An upper bound of S is a number to the right of S in my picture. The existence of cand dare guaranteed by the Least Upper Bound Property. To prove that sis the least upper bound for A, let >0. Another name for a least upper bound is supremum; this is written s= supA: A greatest lower bound or in mum is similarly de ned and is denoted by inf A. Uniqueness of Supremum. (2) Prove that M is the least upper bound for S. Often this is done by assuming that there is an ǫ > 0 such that M − ǫ is also an upper bound for S. One then exhibits an element s ∈ S with s > M − ǫ, showing that M − ǫ is not an upper bound. It is also sometimes called the axiom of Archimedes, although this name is doubly deceptive: it is neither an axiom (it is rather a consequence of the least upper bound property) nor attributed to Archimedes (in fact, Archimedes credits it to Eudoxus). Suppose that A ⊂ R is a set of real numbers. Example 9 Let A = f 1; 1=2; 1=3; 1=4;:::g. Then a least upper bound for A is 0. Below I’ll define what this means, in terms of something called the least upper bound property. Example 8 If A = f1;2;3;:::gthen A has no upper bound, hence no least upper bound. The least upper bound in this last example is actually a maximum for A, that is, an upper bound for A which lies in A. Notallsetshave anupperbound. For example, let’s say you had a set defined by the closed interval [0,2]. The supremum of a set is its least upper bound and the infimum is its greatest upper bound. • Draw a set S of numbers as a subset of the real number line [picture drawn in class]. Exercise 1.4.8. A subset S of R is bounded if it has both an upper bound and a lower bound. The one additional property that we want out of the real numbers is that the real number system should not have any holes in it. Then there exists an n2N so that 1=n< . If m ∈ R is a lower bound …
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