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