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MAT 3749 Introduction to Analysis Section 1.1 The Real Numbers

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Math Party?

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

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

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References Section 1.1 Howland, Section 1.5

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

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

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

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Field

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

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

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Ordered Field

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

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

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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)

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

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

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Example 3 (a) AnalysisSolution

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Example 3 (b) AnalysisSolution

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Example 3 (c) AnalysisSolution

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Archimedean Property

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Density Property

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Least Upper Bounds Similar for greatest lower bound, and infimum

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Equivalent Statement Similar for greatest lower bound, and infimum

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Example 4 Show that AnalysisSolution

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Example 5 Determine the supremum of AnalysisSolution

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The Completeness Axiom

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