1. MAT 3749 Introduction to Analysis Section 1.1 The Real Numbers http://myhome.spu.edu/lauw

2. Math Party?

3. Actuarial Presnentation Cambia/Regence 10/17 3pm OMH 126

4. Preview Field, Ordered Field Lower/Upper Bounds Supremum/ Infimum

5. References Section 1.1 Howland, Section 1.5

6. Introduction: A Story… You are in a foreign country and want to buy….

7. Before going into that,.. We will briefly mention the field properties and that the real numbers R is a field.

8. Before going into that,.. We will briefly mention the field properties and that the real numbers R is a field.

9. Field

10. Real Numbers R The set R of Real Numbers is a field.

11. Example 1 (a) Q is a field. (b) Z is not a field. (Why?)

12. Ordered Field

13. Real Numbers R The set R of Real Numbers is an ordered field.

14. Example 2 C is a field but not an ordered field.

15. Move On… More properties of R. First important milestone of an analysis class – supremum / infimum (Allow us to prove results such as Intermediate Value Theorem)

16. Upper (Lower) Bounds Similar for bounded below, lower bound, and minimum element

17. Example 3 1. Determine which of the following sets are bounded above. 2. Determine which of the following sets have a maximum element.

18. Example 3 (a) AnalysisSolution

19. Example 3 (b) AnalysisSolution

20. Example 3 (c) AnalysisSolution

21. Archimedean Property

22. Density Property

23. Least Upper Bounds Similar for greatest lower bound, and infimum

24. Equivalent Statement Similar for greatest lower bound, and infimum

25. Example 4 Show that AnalysisSolution

26. Example 5 Determine the supremum of AnalysisSolution

27. The Completeness Axiom