MAT 3749 Introduction to Analysis Section 1.1 The Real Numbers

Math Party?

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

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

References Section 1.1 Howland, Section 1.5

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

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

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

Field

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

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

Ordered Field

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

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

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)

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

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

Example 3 (a) AnalysisSolution

Example 3 (b) AnalysisSolution

Example 3 (c) AnalysisSolution

Archimedean Property

Density Property

Least Upper Bounds Similar for greatest lower bound, and infimum

Equivalent Statement Similar for greatest lower bound, and infimum

Example 4 Show that AnalysisSolution

Example 5 Determine the supremum of AnalysisSolution

The Completeness Axiom