1. Introduction to Real Analysis Dr. Weihu Hong Clayton State University 8/21/2008

2. 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?

3. 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.

4. 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.

5. 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 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. Use contradiction to show α²=y Define the real number β by Then If α² α, and by (2), β² y, then by (1), β 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.

11. 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 = -∞