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MAT 3749 Introduction to Analysis Section 1.1 The Real Numbers

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Presentation on theme: "MAT 3749 Introduction to Analysis Section 1.1 The Real Numbers"— Presentation transcript:

1 MAT 3749 Introduction to Analysis Section 1.1 The Real Numbers

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


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